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Use a membership table to show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). 

What is the symbolic word if Caroline is late (l), we will not pursue our plan (~p). We did not pursue our plan. Therefore, Caroline was late.

Show that the square of an even number is an even number using a direct proof.

Show whether or not p←→ q =(p→q) ^ (q→p)

Patrick has assignments in 5 subjects. He can only do two assignments . In how many ways can do two assignments?

Patrick has assignments in 5 subjects he can only do two assignments in how many ways can he do two assignments

How many 5 digit number can be formed from digits 0-6 if:

C. If one (1) is not to be used as the 1st digit and repetition is not allowed

Let f and g be the functions defined by f(x)= 2x+3 g(x)= 3x+2 then composition of f and g is

Using proof by contraposition, show that if n is an integer and 5 added to its cube is odd then n is even. Before showing your solution, rewrite the statement to the proper form of a conditional statement then assign variables to the simple propositions. Show also the contrapositive form of the simple propositions before proceeding to your solution.

Use proof by contradiction to show that a number is even if its square is even. Before showing your solution, rewrite the statement to the proper form of a conditional statement. Assign variables to the simple propositions then write your assumption using these variables and logic symbols. At the end of your solution

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Discrete Mathematics Tutorial

Mathematical logic.

• Propositional Logic
• Discrete Mathematics - Applications of Propositional Logic
• Propositional Equivalences
• Predicates and Quantifiers
• Mathematics | Some theorems on Nested Quantifiers
• Rules of Inference
• Mathematics | Introduction to Proofs

Sets and Relations

• Set Theory - Definition, Types of Sets, Symbols & Examples
• Types Of Sets
• Irreflexive Relation on a Set
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• Types of Functions
• Mathematics | Sequence, Series and Summations
• Representation of Relation in Graphs and Matrices
• What are Relations?
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• Mathematical Induction
• Pigeonhole Principle
• Mathematics | Generalized PnC Set 1
• Discrete Maths | Generating Functions-Introduction and Prerequisites
• Inclusion Exclusion principle and programming applications

Boolean Algebra

• Properties of Boolean Algebra
• Number of Boolean functions
• Minimization of Boolean Functions
• Linear Programming

Ordered Sets & Lattices

• Elements of POSET
• Partial Order Relation on a Set
• Axiomatic Approach to Probability
• Properties of Probability

Probability Theory

• Mathematics | Probability Distributions Set 1 (Uniform Distribution)
• Mathematics | Probability Distributions Set 2 (Exponential Distribution)
• Mathematics | Probability Distributions Set 3 (Normal Distribution)
• Mathematics | Probability Distributions Set 5 (Poisson Distribution)

Graph Theory

• Graph and its representations
• Mathematics | Graph Theory Basics - Set 1
• Types of Graphs with Examples
• Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph
• How to find Shortest Paths from Source to all Vertices using Dijkstra's Algorithm
• Prim’s Algorithm for Minimum Spanning Tree (MST)
• Kruskal’s Minimum Spanning Tree (MST) Algorithm
• Check whether a given graph is Bipartite or not
• Eulerian path and circuit for undirected graph

Special Graph

• Introduction to Graph Coloring
• Edge Coloring of a Graph
• Check if a graph is Strongly, Unilaterally or Weakly connected
• Biconnected Components
• Strongly Connected Components

Group Theory

• Homomorphism & Isomorphism of Group
• Group Isomorphisms and Automorphisms
• Algebraic Structure

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• Last Minute Notes – Discrete Mathematics
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Discrete Mathematics MCQ

• Propositional and First Order Logic.
• Set Theory & Algebra
• Combinatorics
• Top MCQs on Graph Theory in Mathematics
• Numerical Methods and Calculus

Discrete Mathematics is a branch of mathematics that is concerned with “discrete” mathematical structures instead of “continuous”. Discrete mathematical structures include objects with distinct values like graphs, integers, logic-based statements, etc. In this tutorial, we have covered all the topics of Discrete Mathematics for computer science like set theory , recurrence relation, group theory, and graph theory.

Recent Articles on Discrete Mathematics!

• Introduction to Propositional Logic
• Applications of Propositional Logic
• Propositional and Predicate Logic
• Normal and Principle Forms
• Nested Quantifiers Theorem
• Introduction to Proofs
• Types of Sets
• Rough Set Theory
• Sequence and Summations
• Representations of Matrices and Graphs in Relations
• Types of Relation
• Closure of Relation and Equivalence Relations
• Basics of Counting
• Pascal’s Identity
• Permutations and Combinations
• Generalized Permutations and Combinations
• Generating Functions
• Inclusion-Exclusion Principle
• Discrete Probability Theory
• Boolean Functions
• Boolean Algebraic Theorem
• Number of Boolean Functions

Optimization

• Linear Programming 
• Graphical Solution For Linear Programming
• Simplex Algorithm

Ordered Sets & Lattices

• Partially Ordered Sets
• Hasse Diagrams
• Basic Concepts of Probability
• Probability Axioms
• Conditional Probability
• Bayes’ Theorem
• Uniform Distribution
• Exponential Distribution
• Normal Distribution
• Poisson Distribution
• Introduction to Graph
• Basic terminology of a Graph
• Types of a Graph
• Walks, Trails, Paths, and Circuits
• Graph Distance components
• Cut-Vertices and Cut-Edges
• Bridge in Graph
• Independent sets
• Shortest Path Algorithms [Dijkstra’s Algorithm]
• Application of Graph Theory
• Graph Traversals[DFS]
• Graph Traversals[BFS]
• Characterizations of Trees
• Prim’s Minimum Spanning Tree 
• Kruskal’s Minimum Spanning Tree
• Huffman Codes 
• Tree Traversals
• Traveling Salesman Problem
• Bipartite Graphs  
• Independent Sets and Covering
• Eulerian graphs
• Eulerian graphs- Fleury’s algorithm
• Eulerian graphs- Chinese-Postman-Problem Hamilton
• Matching- Basics, Perfect, Bipartite
• Approximation Algorithms

Vertex Colorings

• Chromatic Numbers, Greedy Coloring Algorithm
• Edge Coloring
• Vizing Theorem
• Planar Graph- Basics, Planarity Testing
• Directed Graphs- Degree Centrality
• Directed Graphs- Weak Connectivity
• Directed Graphs- Strong Components 
• Directed Graphs- Eulerian, Hamilton Directed Graphs
• Directed Graphs- Tarjans’ Algorithm To Find Strongly Connected Component
• Handshaking in Graph Theorem
• Groups, Subgroups, Semi Groups
• Isomorphism, Homomorphism
• Automorphism
• Rings, Integral domains, Fields

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Discrete Mathematics: An Open Introduction, 3rd edition

Oscar Levin

Section 1.3 Combinations and Permutations

Investigate.

You have a bunch of chips which come in five different colors: red, blue, green, purple and yellow.

How many different two-chip stacks can you make if the bottom chip must be red or blue? Explain your answer using both the additive and multiplicative principles.

How many different three-chip stacks can you make if the bottom chip must be red or blue and the top chip must be green, purple or yellow? How does this problem relate to the previous one?

How many different three-chip stacks are there in which no color is repeated? What about four-chip stacks?

Suppose you wanted to take three different colored chips and put them in your pocket. How many different choices do you have? What if you wanted four different colored chips? How do these problems relate to the previous one?

A permutation is a (possible) rearrangement of objects. For example, there are 6 permutations of the letters a, b, c :

We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves only 1 choice for the last letter. The multiplicative principle says we multiply \(3\cdot 2 \cdot 1\text{.}\)

Example 1.3.1 .

How many permutations are there of the letters a, b, c, d, e, f ?

We do NOT want to try to list all of these out. However, if we did, we would need to pick a letter to write down first. There are 6 choices for that letter. For each choice of first letter, there are 5 choices for the second letter (we cannot repeat the first letter; we are rearranging letters and only have one of each), and for each of those, there are 4 choices for the third, 3 choices for the fourth, 2 choices for the fifth and finally only 1 choice for the last letter. So there are \(6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\) permutations of the 6 letters.

A piece of notation is helpful here: \(n!\text{,}\) read “\(n\) factorial”, is the product of all positive integers less than or equal to \(n\) (for reasons of convenience, we also define 0! to be 1). So the number of permutation of 6 letters, as seen in the previous example is \(6! = 6\cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\text{.}\) This generalizes:

Permutations of \(n\) elements.

There are \(n! = n\cdot (n-1)\cdot (n-2)\cdot \cdots \cdot 2\cdot 1\) permutations of \(n\) (distinct) elements.

Example 1.3.2 . Counting Bijective Functions.

How many functions \(f:\{1,2,\ldots,8\} \to \{1,2,\ldots, 8\}\) are bijective ?

Remember what it means for a function to be bijective: each element in the codomain must be the image of exactly one element of the domain. Using two-line notation, we could write one of these bijections as

What we are really doing is just rearranging the elements of the codomain, so we are creating a permutation of 8 elements. In fact, “permutation” is another term used to describe bijective functions from a finite set to itself.

If you believe this, then you see the answer must be \(8! = 8 \cdot 7 \cdot\cdots\cdot 1 = 40320\text{.}\) You can see this directly as well: for each element of the domain, we must pick a distinct element of the codomain to map to. There are 8 choices for where to send 1, then 7 choices for where to send 2, and so on. We multiply using the multiplicative principle.

Sometimes we do not want to permute all of the letters/numbers/elements we are given.

Example 1.3.3 .

How many 4 letter “words” can you make from the letters a through f , with no repeated letters?

This is just like the problem of permuting 4 letters, only now we have more choices for each letter. For the first letter, there are 6 choices. For each of those, there are 5 choices for the second letter. Then there are 4 choices for the third letter, and 3 choices for the last letter. The total number of words is \(6\cdot 5\cdot 4 \cdot 3 = 360\text{.}\) This is not \(6!\) because we never multiplied by 2 and 1. We could start with \(6!\) and then cancel the 2 and 1, and thus write \(\frac{6!}{2!}\text{.}\)

In general, we can ask how many permutations exist of \(k\) objects choosing those objects from a larger collection of \(n\) objects. (In the example above, \(k = 4\text{,}\) and \(n = 6\text{.}\)) We write this number \(P(n,k)\) and sometimes call it a \(k\)-permutation of \(n\) elements . From the example above, we see that to compute \(P(n,k)\) we must apply the multiplicative principle to \(k\) numbers, starting with \(n\) and counting backwards. For example

Notice again that \(P(10,4)\) starts out looking like \(10!\text{,}\) but we stop after 7. We can formally account for this “stopping” by dividing away the part of the factorial we do not want:

Careful: The factorial in the denominator is not \(4!\) but rather \((10-4)!\text{.}\)

\(k\)-permutations of \(n\) elements.

\(P(n,k)\) is the number of \(k\)-permutations of \(n\) elements , the number of ways to arrange \(k\) objects chosen from \(n\) distinct objects.

Note that when \(n = k\text{,}\) we have \(P(n,n) = \frac{n!}{(n-n)!} = n!\) (since we defined \(0!\) to be 1). This makes sense —we already know \(n!\) gives the number of permutations of all \(n\) objects.

Example 1.3.4 . Counting injective functions.

How many functions \(f:\{1,2,3\} \to \{1,2,3,4,5,6,7,8\}\) are injective ?

Note that it doesn't make sense to ask for the number of bijections here, as there are none (because the codomain is larger than the domain, there are no surjections). But for a function to be injective, we just can't use an element of the codomain more than once.

We need to pick an element from the codomain to be the image of 1. There are 8 choices. Then we need to pick one of the remaining 7 elements to be the image of 2. Finally, one of the remaining 6 elements must be the image of 3. So the total number of functions is \(8\cdot 7 \cdot 6 = P(8,3)\text{.}\)

What this demonstrates in general is that the number of injections \(f:A \to B\text{,}\) where \(\card{A} = k\) and \(\card{B} = n\text{,}\) is \(P(n,k)\text{.}\)

Here is another way to find the number of \(k\)-permutations of \(n\) elements: first select which \(k\) elements will be in the permutation, then count how many ways there are to arrange them. Once you have selected the \(k\) objects, we know there are \(k!\) ways to arrange (permute) them. But how do you select \(k\) objects from the \(n\text{?}\) You have \(n\) objects, and you need to choose \(k\) of them. You can do that in \({n \choose k}\) ways. Then for each choice of those \(k\) elements, we can permute them in \(k!\) ways. Using the multiplicative principle, we get another formula for \(P(n,k)\text{:}\)

Now since we have a closed formula for \(P(n,k)\) already, we can substitute that in:

If we divide both sides by \(k!\) we get a closed formula for \({n \choose k}\text{.}\)

Closed formula for \({n \choose k}\).

We say \(P(n,k)\) counts permutations , and \({n \choose k}\) counts combinations . The formulas for each are very similar, there is just an extra \(k!\) in the denominator of \({n \choose k}\text{.}\) That extra \(k!\) accounts for the fact that \({n \choose k}\) does not distinguish between the different orders that the \(k\) objects can appear in. We are just selecting (or choosing) the \(k\) objects, not arranging them. Perhaps “combination” is a misleading label. We don't mean it like a combination lock (where the order would definitely matter). Perhaps a better metaphor is a combination of flavors — you just need to decide which flavors to combine, not the order in which to combine them.

To further illustrate the connection between combinations and permutations, we close with an example.

Example 1.3.5 .

You decide to have a dinner party. Even though you are incredibly popular and have 14 different friends, you only have enough chairs to invite 6 of them.

How many choices do you have for which 6 friends to invite?

What if you need to decide not only which friends to invite but also where to seat them along your long table? How many choices do you have then?

You must simply choose 6 friends from a group of 14. This can be done in \({14 \choose 6}\) ways. We can find this number either by using Pascal's triangle or the closed formula: \(\frac{14!}{8!\cdot 6!} = 3003\text{.}\)

Here you must count all the ways you can permute 6 friends chosen from a group of 14. So the answer is \(P(14, 6)\text{,}\) which can be calculated as \(\frac{14!}{8!} = 2162160\text{.}\)

Notice that we can think of this counting problem as a question about counting functions: how many injective functions are there from your set of 6 chairs to your set of 14 friends (the functions are injective because you can't have a single chair go to two of your friends).

How are these numbers related? Notice that \(P(14,6)\) is much larger than \({14 \choose 6}\text{.}\) This makes sense. \({14 \choose 6}\) picks 6 friends, but \(P(14,6)\) arranges the 6 friends as well as picks them. In fact, we can say exactly how much larger \(P(14,6)\) is. In both counting problems we choose 6 out of 14 friends. For the first one, we stop there, at 3003 ways. But for the second counting problem, each of those 3003 choices of 6 friends can be arranged in exactly \(6!\) ways. So now we have \(3003\cdot 6!\) choices and that is exactly \(2162160\text{.}\)

Alternatively, look at the first problem another way. We want to select 6 out of 14 friends, but we do not care about the order they are selected in. To select 6 out of 14 friends, we might try this:

This is a reasonable guess, since we have 14 choices for the first guest, then 13 for the second, and so on. But the guess is wrong (in fact, that product is exactly \(2162160 = P(14,6)\)). It distinguishes between the different orders in which we could invite the guests. To correct for this, we could divide by the number of different arrangements of the 6 guests (so that all of these would count as just one outcome). There are precisely \(6!\) ways to arrange 6 guests, so the correct answer to the first question is

Note that another way to write this is

which is what we had originally.

Exercises Exercises

In an attempt to clean up your room, you have purchased a new floating shelf to put some of your 17 books you have stacked in a corner. These books are all by different authors. The new book shelf is large enough to hold 10 of the books.

How many ways can you select and arrange 10 of the 17 books on the shelf? Notice that here we will allow the books to end up in any order. Explain.

How many ways can you arrange 10 of the 17 books on the shelf if you insist they must be arranged alphabetically by author? Explain.

Which question should have the larger answer? One of these is a combination, the other is a permutation.

Suppose you wanted to draw a quadrilateral using the dots below as vertices (corners). The dots are spaced one unit apart horizontally and two units apart vertically.

How many triangles are there with vertices from the points shown below? Note, we are not allowing degenerate triangles - ones with all three vertices on the same line, but we do allow non-right triangles. Explain why your answer is correct.

If you pick any three points, you can get a triangle, unless those three points are all on the \(x\)-axis or on the \(y\)-axis. There are other ways to start this as well, and any correct method should give the same answer.

We have seen that the formula for \(P(n,k)\) is \(\dfrac{n!}{(n-k)!}\text{.}\) Your task here is to explain why this is the right formula.

Suppose you have 12 chips, each a different color. How many different stacks of 5 chips can you make? Explain your answer and why it is the same as using the formula for \(P(12,5)\text{.}\)

Using the scenario of the 12 chips again, what does \(12!\) count? What does \(7!\) count? Explain.

Explain why it makes sense to divide \(12!\) by \(7!\) when computing \(P(12,5)\) (in terms of the chips).

Does your explanation work for numbers other than 12 and 5? Explain the formula \(P(n,k) = \frac{n!}{(n-k)!}\) using the variables \(n\) and \(k\text{.}\)

Browse Course Material

Course info, instructors.

• Prof. Ronitt Rubinfeld
• Prof. Albert R. Meyer

Departments

• Electrical Engineering and Computer Science
• Mathematics

As Taught In

• Computer Science
• Applied Mathematics
• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Mathematics for computer science, course description.

This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds:

• Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations
• Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting
• Discrete Probability Theory

A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science).

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• Probability in Discrete Math: Tackling Assignment Challenges

The Role of Probability in Discrete Math: Tackling Assignment Challenges

Discrete Mathematics, often hailed as the backbone of computer science, encompasses various mathematical structures and techniques that deal with distinct, separate values, providing the theoretical foundation for solving real-world problems in computer science, cryptography, and information theory. Among the many facets of discrete math, probability theory emerges as a linchpin, playing a pivotal role in understanding and solving complex problems that involve uncertainty and randomness. In this comprehensive blog post, we will delve into the significance of probability in discrete mathematics, unraveling its intricate connections with combinatorics, graph theory, and real-world applications. Specifically focusing on its application in tackling assignment challenges, we will explore how probability becomes an indispensable tool in optimizing resource allocation, modeling uncertainties, and making informed decisions in various domains, from logistics to project management. As we navigate through the fundamental concepts of probability, combinatorics, and graph theory, we will unravel the symbiotic relationship between these branches of discrete mathematics and showcase how probability becomes the thread weaving through the tapestry of problem-solving. From understanding sample spaces and events to mastering the multiplication principle and combinations, readers will gain insights into the mathematical toolkit that empowers them to analyze and solve intricate problems in discrete mathematics. If you need assistance with your Discrete Math assignment , this comprehensive exploration can provide valuable insights into the subject matter, aiding in your understanding and completion of assignments in this field.

Moreover, we will delve into the application of probability in graph theory, where random walks and Markov chains become instrumental in modeling dynamic processes and predicting system behavior. Transitioning from theory to practicality, we will present a case study in project management, illustrating how probabilistic assignment models can optimize resource allocation, considering uncertainties in task durations. However, as we embrace the power of probability in discrete mathematics, we cannot overlook the challenges and considerations that come with it. Data quality and accuracy, computational complexity, and the need for continuous refinement of models pose significant hurdles in effectively applying probability to solve real-world problems. Nevertheless, the ever-evolving landscape of technology and computational advancements continues to pave the way for innovative solutions to these challenges. In conclusion, this exploration into the role of probability in discrete mathematics reveals its transformative impact on problem-solving, decision-making, and modeling uncertainties. As technology advances, the integration of probability into discrete math models will remain at the forefront, shaping the future of computer science and its applications across diverse domains.

Understanding Discrete Mathematics

Understanding Discrete Mathematics is foundational to navigating the intricate landscapes of computer science and problem-solving. Unlike continuous mathematics, which deals with smooth and unbroken values, discrete mathematics focuses on distinct, countable entities. This branch encompasses a diverse array of mathematical structures and techniques, including set theory, logic, combinatorics, graph theory, and, notably, probability theory. Discrete mathematics serves as the bedrock for solving real-world problems in computer science, cryptography, and information theory. At its core, the discipline addresses the nature of countable sets, offering a framework for examining objects that can be distinctly enumerated. Concepts such as sample spaces, events, and probability distributions become crucial as we explore the probabilistic dimension within discrete mathematics, paving the way for a deeper understanding of uncertainty and randomness. The foundational principles laid out in discrete mathematics provide the necessary tools for approaching complex problems systematically, making it an indispensable field for those seeking to unravel the intricacies of computation and logical reasoning.

1. Countable Sets and Enumeration:

Discrete mathematics delves into the nature of countable sets, providing a framework for the systematic enumeration of objects. Whether analyzing the elements of a set or understanding the combinations and permutations within a given context, the ability to count and enumerate forms the basis for many problem-solving strategies in discrete mathematics.

2. Probabilistic Dimension in Discrete Mathematics:

The incorporation of probability theory within discrete mathematics introduces a probabilistic dimension, enabling a deeper understanding of uncertainty and randomness. Concepts such as sample spaces, events, and probability distributions become essential tools, offering a structured approach to analyze uncertain situations and make informed decisions within the discrete realm. This probabilistic perspective not only enriches theoretical foundations but also enhances the applicability of discrete mathematics in addressing real-world challenges in computer science and beyond.

The Fundamental Concepts of Probability

The fundamental concepts of probability form the bedrock of this mathematical discipline, serving as a guiding light in navigating uncertain scenarios. At its core, probability theory revolves around two key elements: sample spaces and events. The sample space encapsulates all conceivable outcomes of an experiment, while events represent subsets within this space. This conceptual foundation sets the stage for a nuanced understanding of likelihood and uncertainty. Within this framework, probability distributions emerge as a crucial tool, assigning probabilities to each possible outcome. Whether uniform or non-uniform, these distributions become instrumental in quantifying the likelihood of events. Combinatorics, another integral facet, intersects with probability through principles like the multiplication principle and combinations. The multiplication principle delineates the myriad ways independent events can unfold, a key insight when addressing complex scenarios. Combinations, on the other hand, prove indispensable in counting problems where the order of selection is irrelevant. These fundamental concepts collectively lay the groundwork for a profound exploration of probability's role in discrete mathematics, fostering a comprehensive understanding of uncertain situations and contribu

1. Sample Spaces and Events

A sample space represents the set of all possible outcomes of an experiment, and an event is a subset of the sample space. For instance, consider rolling a six-sided die. The sample space would be {1, 2, 3, 4, 5, 6}, and an event could be rolling an even number, represented by the set {2, 4, 6}.

2. Probability Distributions

Probability distributions assign probabilities to each possible outcome of an experiment. In discrete mathematics, probability distributions can be either uniform or non-uniform. The uniform distribution gives equal probability to all outcomes, while the non-uniform distribution assigns different probabilities to different outcomes.

Probability in Combinatorics

Probability plays a crucial role in combinatorics, a fundamental branch of discrete mathematics focused on counting and arranging objects. Within combinatorics, understanding the probability of various outcomes is essential for solving problems related to combinations, permutations, and other counting scenarios. The multiplication principle, a fundamental concept, relies on probability when dealing with independent events. By assigning probabilities to individual outcomes, combinatorics gains a nuanced perspective, enabling the analysis of complex scenarios where the likelihood of different combinations varies. Combinations, often denoted as C(n, k), represent a key application of probability in combinatorics. They quantify the number of ways to choose k elements from a set of n elements without considering the order, emphasizing the significance of probability in scenarios where the arrangement is not a primary concern. In essence, the integration of probability in combinatorics enriches the field, providing a more comprehensive toolkit for solving problems that involve counting, arranging, and selecting elements from finite sets. This interplay between probability and combinatorics is pivotal in addressing real-world problems across diverse domains, further highlighting the interconnectedness of these mathematical concepts.

1. The Multiplication Principle

The multiplication principle states that if there are m ways to do one thing and n ways to do another, then there are m * n ways to do both. This principle is particularly useful when considering independent events. For example, if you have 3 choices for breakfast and 4 choices for lunch, there are 3 * 4 = 12 different ways to choose both meals.

2. Combinations

Combinations are a way to count the number of ways to choose k elements from a set of n elements without regard to the order. The formula for combinations, denoted as C(n, k), is essential in solving problems where the order of selection doesn't matter. Understanding combinations is crucial in various probability scenarios, such as selecting a committee or forming a group of students.

Probability in Graph Theory

Probability in Graph Theory is a vital aspect that brings a stochastic dimension to the study of graphs and their dynamic processes. Graphs, representing interconnected nodes and edges, find applications in diverse fields, and understanding the probabilistic behavior within them is essential. Random walks on graphs exemplify this, involving the traversal from one vertex to another based on probabilistic transitions. This concept is fundamental in modeling scenarios like the spread of information in social networks or the movement of particles in physical systems. Moreover, Markov Chains, a cornerstone of probability in graph theory, encapsulate stochastic models where transition probabilities depend solely on the current state, offering insights into the long-term behavior of processes. The probabilistic lens in graph theory not only enhances the theoretical framework but also finds practical applications in algorithmic analysis and network dynamics. Whether predicting the diffusion of influence in a network or modeling random processes on interconnected structures, probability in graph theory emerges as a crucial tool for comprehending uncertainties within complex systems and optimizing decision-making processes.

1. Random Walks on Graphs

Random walks on graphs involve moving from one vertex to another randomly based on certain probabilities. These random processes have applications in modeling various real-world scenarios, such as the spread of information on social networks or the movement of particles in a physical system. Probability provides the tools to analyze and predict the behavior of random walks on graphs.

2. Markov Chains

Markov chains are stochastic models that describe a sequence of events where the probability of transitioning to a particular state depends solely on the current state and time elapsed. These chains are extensively used in graph theory and discrete mathematics to model processes with probabilistic transitions. Understanding the probabilities associated with state transitions is crucial in predicting the long-term behavior of Markov chains.

Probability in Assignment Challenges

Probability takes center stage in addressing assignment challenges within the realm of discrete mathematics. Assignment problems, pervasive in logistics, operations research, and scheduling, necessitate optimal resource allocation to tasks or individuals while considering uncertainties. The integration of probability introduces a dynamic layer to assignment models, particularly when dealing with variable task durations or fluctuating resource availabilities. By leveraging probabilistic models, decision-makers gain a nuanced understanding of the potential variability in project completion times. This becomes especially crucial in scenarios where task durations follow probability distributions, such as the normal distribution. A compelling case study in project management exemplifies the practical application of probability, where tasks with probabilistic durations are strategically assigned resources to minimize the expected project completion time. The optimization process involves not only considering the most likely durations but the entire probability distributions, employing techniques like Monte Carlo simulation to simulate diverse project scenarios. As probability becomes an integral component in addressing assignment challenges, the field continues to grapple with considerations like data accuracy and computational complexity, emphasizing the need for reliable data and efficient algorithms to ensure the success of probabilistic assignment models.

1. Assignment Problems in Discrete Mathematics

Assignment problems involve the optimal allocation of resources to tasks or individuals, with the objective of minimizing or maximizing a certain criterion. These problems are prevalent in various domains, including logistics, operations research, and scheduling. Probability comes into play when uncertainties are present in the assignment process, such as variable task durations or fluctuating resource availability.

2. Probabilistic Models for Assignment Problems

Probabilistic models are indispensable when dealing with uncertainties in assignment problems. These models consider the probabilistic nature of task durations, resource availabilities, or other relevant factors. By incorporating probability distributions into assignment models, decision-makers can make more informed and robust decisions.

Case Study: Probabilistic Assignment in Project Management

In a practical demonstration of the pivotal role probability plays in discrete mathematics, let's delve into a case study within the realm of project management. Consider a complex project comprising multiple tasks, each characterized by a probability distribution representing task durations. The fundamental idea is to optimize resource allocation with the overarching goal of minimizing the expected project completion time. Modeling task durations using probability distributions, such as the normal distribution, enables project managers to account for uncertainties inherent in real-world scenarios. The optimization process involves not only considering the most likely task durations but the entire probability distribution, incorporating nuances like variability and risk. Employing techniques like Monte Carlo simulation allows for the generation of multiple project scenarios, each reflecting different random realizations of task durations. This comprehensive approach empowers decision-makers to assess the likelihood of meeting project deadlines under various resource allocation strategies, showcasing the practical relevance and effectiveness of probability in addressing intricate assignment challenges in the dynamic landscape of project management.

1. Modeling Task Durations with Probability Distributions

In a probabilistic assignment scenario, each task's duration is modeled using a probability distribution, such as the normal distribution. By characterizing the uncertainty in task durations, project managers can gain insights into the potential variability in project completion times.

2. Optimizing Resource Allocation with Probability

Optimizing resource allocation involves considering not only the most likely duration of each task but also the entire probability distribution. Techniques such as Monte Carlo simulation can be employed to simulate multiple project scenarios, each with different random realizations of task durations. This simulation approach allows decision-makers to assess the likelihood of meeting project deadlines under various resource allocation strategies.

Challenges and Considerations in Probabilistic Assignment

Despite the undeniable advantages of employing probability in tackling assignment challenges within the realm of discrete mathematics, several challenges and considerations warrant careful attention. One prominent concern revolves around the quality and accuracy of the data underpinning probabilistic models. The efficacy of these models hinges on reliable input data, and inaccuracies in probability distributions or task duration estimates may lead to suboptimal assignment decisions. Consequently, ensuring the acquisition of precise data and implementing mechanisms for continuous model refinement based on real-world feedback emerges as a critical imperative. Another substantial challenge lies in the computational complexity associated with probabilistic assignment problems, particularly in scenarios involving intricate probability distributions and large-scale systems. The efficient resolution of such complexities demands the development of sophisticated algorithms and computational techniques. Striking a balance between the need for precision in probabilistic modeling and the computational efficiency required for practical implementation remains an ongoing challenge. Addressing these challenges will be instrumental in fortifying the applicability and reliability of probabilistic assignment solutions across diverse domains, fostering a more resilient and accurate decision-making framework in the face of uncertainty.

1. Data Quality and Accuracy

The effectiveness of probabilistic models heavily relies on the quality and accuracy of the input data. Inaccurate probability distributions or unreliable estimates of task durations can lead to suboptimal assignment decisions. Therefore, obtaining reliable data and continuously updating models based on real-world feedback are critical aspects of successful probabilistic assignment.

2. Computational Complexity

Probabilistic assignment problems, especially those involving complex probability distributions and large-scale systems, can pose computational challenges. Efficient algorithms and computational techniques are necessary to handle the increased complexity associated with probabilistic models. Researchers and practitioners in the field continually work towards developing scalable solutions for such challenges.

In conclusion, probability plays a pivotal role in discrete mathematics, offering a powerful framework for addressing assignment challenges in various domains. From combinatorics to graph theory and project management, the application of probability enhances decision-making processes and enables a deeper understanding of uncertain scenarios. As technology advances and computational tools become more sophisticated, the integration of probability into discrete math models will continue to evolve, providing innovative solutions to complex assignment problems. Embracing the probabilistic perspective in discrete mathematics not only enriches theoretical foundations but also empowers practitioners to make more informed and robust decisions in the face of uncertainty.

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April 2024 NPTEL Exams - Hall Tickets Released!

Discrete Mathematics: Assignment 11 Reevaluations!!

Dear Learners, Assignment 11 submission of all students has been reevaluated by updating the answer for Question 10. Students are requested to find their revised scores of Assignment 11 on the Progress page. -NPTEL Team

Discrete Mathematics: Assignment 6 Reevaluations!!

Dear Learners, Assignment 6 submission of all students has been reevaluated by changing the answer for Question 9. Students are requested to find their revised scores of Assignment 6 on the Progress page. -NPTEL Team

Discrete Mathematics : Assignment 8 Reevaluations!!

Dear Learners, Assignment 8 submission of all students has been reevaluated by changing the option for Question 4. Students are requested to find their revised scores of Assignment 8 on the Progress page. -NPTEL Team

Exam Format - April, 2024!!

Dear Candidate, ****This is applicable only for the exam registered candidates**** Type of exam will be available in the list: Click Here You will have to appear at the allotted exam center and produce your Hall ticket and Government Photo Identification Card (Example: Driving License, Passport, PAN card, Voter ID, Aadhaar-ID with your Name, date of birth, photograph and signature) for verification and take the exam in person.  You can find the final allotted exam center details in the hall ticket. The hall ticket is yet to be released.  We will notify the same through email and SMS. Type of exam: Computer based exam (Please check in the above list corresponding to your course name) The questions will be on the computer and the answers will have to be entered on the computer; type of questions may include multiple choice questions, fill in the blanks, essay-type answers, etc. Type of exam: Paper and pen Exam  (Please check in the above list corresponding to your course name) The questions will be on the computer. You will have to write your answers on sheets of paper and submit the answer sheets. Papers will be sent to the faculty for evaluation. On-Screen Calculator Demo Link: Kindly use the below link to get an idea of how the On-screen calculator will work during the exam. https://tcsion.com/ OnlineAssessment/ ScientificCalculator/ Calculator.html NOTE: Physical calculators are not allowed inside the exam hall. Thank you! -NPTEL Team

Discrete Mathematics : Assignment 11 Solutions Released!!

Dear Learners, The Solutions of Week 11 for the course "Discrete Mathematics " have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 11 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/pdf/noc24_cs58/Week%2011%20(1).pdf Happy Learning! Thanks & Regards, NPTEL Team

Discrete Mathematics : Assignment 10 Solutions Released!!

Dear Learners, The Solutions of Week 10 for the course " Discrete Mathematics " have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 10 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/pdf/noc24_cs58/Week%2010%20(1).pdf Happy Learning! Thanks & Regards, NPTEL Team

Week 12 Feedback Form: Discrete Mathematics!!

Dear Learners, Thank you for continuing with the course and hope you are enjoying it. We would like to know if the expectations with which you joined this course are being met and hence please do take 2 minutes to fill out our weekly feedback form. It would help us tremendously in gauging the learner experience. Here is the link to the form:    https://docs.google.com/forms/d/1hvOiA9Krfolm54yTBWpu8c1Vj0fEMuqqPeIe33C34Lc/viewform Thank you -NPTEL team

Discrete Mathematics :Week 12 content & Assignment live now!!

Dear Learners, The lecture videos for Week 12 have been uploaded for the course "Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=491&lesson=492 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-12 for Week-12 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=491&assessment=597 The assignment has to be submitted on or before Wednesday,[17/04/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Discrete Mathematics :Week 11 content & Assignment live now!!

Dear Learners, The lecture videos for Week 11 have been uploaded for the course " Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=456&lesson=457 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-11 for Week-11 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=456&assessment=602 The assignment has to be submitted on or before Wednesday,[10/04/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Discrete Mathematics : Assignment 7, 8 and 9 Solutions Released!!

Dear Learners, The Solutions of Week 7, 8 and Week 9 for the course "Discrete Mathematics" have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 7 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/pdf/noc24_cs58/Week%207.pdf Assignment 8 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/pdf/noc24_cs58/Week%208.pdf Assignment 9 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/img/noc24_cs58/Week%209.pdf Happy Learning! Thanks & Regards, NPTEL Team

Important Notice:No CHANGE in NPTEL Exam Schedule for April 2024

Dear Student,

We wanted to take a moment to address an important matter regarding the upcoming election dates and their potential impact on your exam schedule.

• None of the election dates clash with scheduled exam dates. If we schedule additional dates, we will ensure they again do not clash with elections in your state. 
• Hence this is to confirm that there will be no changes to the exam dates and they are the same as previously scheduled. We may have exams in some cities on April 19 and April 26 depending on seat availability on scheduled dates. But again this will be done ensuring we don't conduct exams on election dates in your state. 
• Your academic progress and success remain our top priority, and we are committed to maintaining the integrity of the examination process.
• We have more than 6 lakh learners registered for April exams and logistics has been a huge challenge. We understand that some of you may need to travel to your native cities to participate in the voting process. Please remember that you selected your exam cities during registration, and it is crucial that you return to these cities to take your exams as scheduled. Since hall ticket and center allocation is under process, exam cities selected by you during exam registration cannot be changed now. 

Hence we kindly request that you make the necessary arrangements to ensure you can both exercise your right to vote and fulfill your academic obligations.

Warm Regards,

NPTEL Team.

Week 10 Feedback Form: Discrete Mathematics!!

Discrete mathematics :week 10 content & assignment live now.

Dear Learners, The lecture videos for Week 10 have been uploaded for the course " Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=423&lesson=424 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-10 for Week-10 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=423&assessment=601 The assignment has to be submitted on or before Wednesday,[03/04/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Week 9 Feedback Form: Discrete Mathematics!!

Discrete mathematics :week 9 content & assignment live now.

Dear Learners, The lecture videos for Week 9 have been uploaded for the course " Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=390&lesson=391 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-9 for Week-9 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=390&assessment=600 The assignment has to be submitted on or before Wednesday,[27/03/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Week 8 Feedback Form: Discrete Mathematics!!

Discrete mathematics :week 8 content & assignment live now.

Dear Learners, The lecture videos for Week 8 have been uploaded for the course "Discrete Mathematics" . The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=330&lesson=331 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-8 for Week-8 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=330&assessment=599 The assignment has to be submitted on or before Wednesday,[20/03/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Discrete Mathematics : Assignment 5 and 6 Solutions Released!!

Dear Learners, The Solutions of Week 5 and Week 6 for the course " Discrete Mathematics" have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 5 Solution Link:   https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/img/noc24_cs58/Week%205%20(1).pdf Assignment 6 Solution Link:  https://storage.googleapis.com/swayam-node1-production.appspot.com/assets/img/noc24_cs58/Week%206%20(1).pdf Happy Learning! Thanks & Regards, NPTEL Team

Week 7 Feedback Form: Discrete Mathematics!!

Discrete mathematics :week 7 content & assignment live now.

Dear Learners, The lecture videos for Week 7 have been uploaded for the course "Discrete Mathematics" . The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=284&lesson=285 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-7 for Week-7 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=284&assessment=598 The assignment has to be submitted on or before Wednesday,[13/03/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Week 6 Feedback Form: Discrete Mathematics!!

Discrete mathematics: assignment 3 and 4 solutions released.

Dear Learners, The Solutions of Week 3 and Week 4 for the course "Discrete Mathematics" have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 3 Solution Link:  https://drive.google.com/file/d/1r4vk6NJVi43OkxnecrGtE5y_Knem1qtZ/view Assignment 4 Solution Link:  https://drive.google.com/file/d/18dnikR-A_Z9F-17ck3NBBs1NvZB6MAhy/view Happy Learning! Thanks & Regards, NPTEL Team

Discrete Mathematics :Week 6 content & Assignment live now!!

Dear Learners, The lecture videos for Week 6 have been uploaded for the course " Discrete Mathematics " . The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=242&lesson=243 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-6 for Week-6 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=242&assessment=596 The assignment has to be submitted on or before Wednesday,[06/03/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

NPTEL: Exam Registration date is extended for 12 week courses of Jan 2024!

• No further extension will be provided.
• This extension is only applicable for 12-week courses.

Discrete Mathematics : Assignment 3 Reevaluations!!

Dear Learners, Assignment 3 submission of all students has been reevaluated by updating the answer Question 1. Students are requested to find their revised scores of Assignment 3 on the Progress page. -NPTEL Team

Reminder: NPTEL: Exam Registration is date is extended for Jan 2024 courses!

Dear Learner,  The exam registration for the Jan 2024 NPTEL course certification exam is extended till February 23, 2024 - 05.00 P.M . CLICK HERE to register for the exam Choose from the Cities where exam will be conducted: Exam Cities Click here to view Timeline and Guideline : Guideline For further details on registration process please refer the previous announcement in the course page. -NPTEL Team

Week 5 Feedback Form: Discrete Mathematics!!

Dear Learner,  The exam registration for the Jan 2024 NPTEL course certification exam is extended till February 20, 2024 - 05.00 P.M . CLICK HERE to register for the exam Choose from the Cities where exam will be conducted: Exam Cities Click here to view Timeline and Guideline : Guideline For further details on registration process please refer the previous announcement in the course page. -NPTEL Team

Discrete Mathematics :Week 5 content & Assignment live now!!

Dear Learners, The lecture videos for Week 5 have been uploaded for the course "Discrete Mathematics". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=192&lesson=193 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-5 for Week-5 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=192&assessment=595 The assignment has to be submitted on or before Wednesday,[28/02/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Week 4 Feedback Form: Discrete Mathematics!!

Discrete mathematics: assignment 1 and 2 solutions released.

Dear Learners, The Solutions of Week 1 and Week 2 for the course " Discrete Mathematics " have been released in the portal. Please go through the solution and in case of any doubt post your queries in the forum. Assignment 1 Solution Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=18&lesson=530 Assignment 2 Solution Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=60&lesson=531 Happy Learning! Thanks & Regards, NPTEL Team

Discrete Mathematics: Week 4 content & Assignment is live now !!

Dear Learners, The lecture videos for Week 04 have been uploaded for the course " Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=145&lesson=146 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-4 for Week-4 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=145&assessment=594 The assignment has to be submitted on or before Wednesday,[21/02/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Discrete Mathematics: Reminder for Assignment 1 & 2 deadline!!

Dear Learners, The Deadline for Assignments 1 & 2 will close on Wednesday,[07/02/2024], 23:59 IST. Kindly submit the assignments before the deadline. Thanks and Regards, -NPTEL Team

Week 3 Feedback Form: Discrete Mathematics!!

Discrete mathematics: week 3 content & assignment is live now .

Dear Learners, The lecture videos for Week 3 have been uploaded for the course " Discrete Mathematics " . The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=92&lesson=93 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-3 for Week-3 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=92&assessment=593 The assignment has to be submitted on or before Wednesday,[14/02/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Week 2 Feedback Form: Discrete Mathematics!!

Discrete mathematics: week 2 content & assignment is live now .

Dear Learners, The lecture videos for Week 2 have been uploaded for the course " Discrete Mathematics" . The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=60&lesson=61 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-2 for Week-2 is also released and can be accessed from the following link Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=60&assessment=592 The assignment has to be submitted on or before Wednesday,[07/02/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Reminder: NPTEL: Exam Registration is open now for Jan 2024 courses!

Dear Learner, 

Here is the much-awaited announcement on registering for the Jan 2024 NPTEL course certification exam. 

1. The registration for the certification exam is open only to those learners who have enrolled in the course. 

2. If you want to register for the exam for this course, login here using the same email id which you had used to enroll to the course in Swayam portal. Please note that Assignments submitted through the exam registered email id ALONE will be taken into consideration towards final consolidated score & certification. 

3 . Date of exam: Apr 28, 2024 

CLICK HERE to register for the exam.

Choose from the Cities where exam will be conducted: Exam Cities

4. Exam fees: 

If you register for the exam and pay before Feb 12, 2024 - 5:00 PM, Exam fees will be Rs. 1000/- per exam .

5. 50% fee waiver for the following categories: 

Students belonging to the SC/ST category: please select Yes for the SC/ST option and upload the correct Community certificate.

Students belonging to the PwD category with more than 40% disability: please select Yes for the option and upload the relevant Disability certificate. 

6. Last date for exam registration: Feb 16, 2024 - 5:00 PM (Friday). 

7. Between Feb 12, 2024 - 5:00 PM & Feb 16, 2024 - 5:00 PM late fee will be applicable.

8. Mode of payment: Online payment - debit card/credit card/net banking/UPI. 

9. HALL TICKET: 

The hall ticket will be available for download tentatively by 2 weeks prior to the exam date. We will confirm the same through an announcement once it is published. 

10. FOR CANDIDATES WHO WOULD LIKE TO WRITE MORE THAN 1 COURSE EXAM:- you can add or delete courses and pay separately – till the date when the exam form closes. Same day of exam – you can write exams for 2 courses in the 2 sessions. Same exam center will be allocated for both the sessions. 

11. Data changes: 

Last date for data changes: Feb 16, 2024 - 5:00 PM :  

We will charge an additional fee of Rs. 200 to make any changes related to name, DOB, photo, signature, SC/ST and PWD certificates after the last date of data changes.

The following 6 fields can be changed (until the form closes) ONLY when there are NO courses in the course cart. And you will be able to edit those fields only if you: - 

REMOVE unpaid courses from the cart And/or - CANCEL paid courses 

1. Do you come under the SC/ST category? * 

2. SC/ST Proof 

3. Are you a person with disabilities? * 

4. Are you a person with disabilities above 40%? 

5. Disabilities Proof 

6. What is your role? 

Note: Once you remove or cancel a course, you will be able to edit these fields immediately. 

But, for cancelled courses, refund of fees will be initiated only after 2 weeks. 

12. LAST DATE FOR CANCELLING EXAMS and getting a refund: Feb 16, 2024 - 5:00 PM  

13. Click here to view Timeline and Guideline : Guideline

Domain Certification

Domain Certification helps learners to gain expertise in a specific Area/Domain. This can be helpful for learners who wish to work in a particular area as part of their job or research or for those appearing for some competitive exam or becoming job ready or specialising in an area of study.  

Every domain will comprise Core courses and Elective courses. Once a learner completes the requisite courses as per the mentioned criteria, you will receive a Domain Certificate showcasing your scores and the domain of expertise. Kindly refer to the following link for the list of courses available under each domain: https://nptel.ac.in/domains

Outside India Candidates

Candidates who are residing outside India may also fill the exam form and pay the fees. Mode of exam and other details will be communicated to you separately.

Thanks & Regards, 

Week 1 Feedback Form: Discrete Mathematics!!

Discrete mathematics: week 1 content & assignment is live now .

Dear Learners, The lecture videos for Week 1 have been uploaded for the course " Discrete Mathematics ". The lectures can be accessed using the following link: Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=18&lesson=19 The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already). Assignment-1 for Week-1 is also released and can be accessed from the following link Link: https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=18&assessment=591 The assignment has to be submitted on or before Wednesday,[07/02/2024], 23:59 IST. As we have done so far, please use the discussion forums if you have any questions on this module. Note : Please check the due date of the assignments in the announcement and assignment page if you see any mismatch write to us immediately. Thanks and Regards, -NPTEL Team

Discrete Mathematics : Week-1 video is live now !!

Dear Learners,

The lecture videos for Week 1 have been uploaded for the course “ Discrete Mathematics ”. The lectures can be accessed using the following link.

Link:  https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=18&lesson=19

The other lectures in this week are accessible from the navigation bar to the left. Please remember to login into the website to view contents (if you aren't logged in already).

Assignment will be released shortly.

As we have done so far, please use the discussion forums if you have any questions on this module.

Thanks and Regards,

-NPTEL Team

Discrete Mathematics : Assignment 0 is live now!!

Dear Learners, We welcome you all to this course " Discrete Mathematics ". The assignment 0 has been released. This assignment is based on a prerequisite of the course. You can find the assignment in the link: https://onlinecourses.nptel.ac.in/noc24_cs58/unit?unit=16&assessment=590 Please note that this assignment is for practice and it will not be graded. Thanks & Regards   -NPTEL Team

NPTEL: Exam Registration is open now for Jan 2024 courses!

Discrete mathematics: welcome to nptel online course - jan 2024.

• Every week, about 2.5 to 4 hours of videos containing content by the Course instructor will be released along with an assignment based on this. Please watch the lectures, follow the course regularly and submit all assessments and assignments before the due date. Your regular participation is vital for learning and doing well in the course. This will be done week on week through the duration of the course.
• Please do the assignments yourself and even if you take help, kindly try to learn from it. These assignments will help you prepare for the final exams. Plagiarism and violating the Honor Code will be taken very seriously if detected during the submission of assignments.
• The announcement group - will only have messages from course instructors and teaching assistants - regarding the lessons, assignments, exam registration, hall tickets, etc.
• The discussion forum (Ask a question tab on the portal) - is for everyone to ask questions and interact. Anyone who knows the answers can reply to anyone's post and the course instructor/TA will also respond to your queries.
• Please make maximum use of this feature as this will help you learn much better.
• If you have any questions regarding the exam, registration, hall tickets, results, queries related to the technical content in the lectures, any doubts in the assignments, etc can be posted in the forum section
• The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
• The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
• Date and Time of Exams: April 28, 2024 Morning session 9am to 12 noon; Afternoon Session 2 pm to 5 pm.
• Registration URL: Announcements will be made when the registration form is open for registrations.
• The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
• Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.
• Once again, thanks for your interest in our online courses and certification. Happy learning.

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