Discrete Mathematical Structures (Counting Principles)

Discrete Mathematical Structures(Counting Principles)

Discrete Mathematical Structures: Theory and Applications 2

Learning Objectives

Learn the basic counting principles—multiplication and addition

Explore the pigeonhole principle

Learn about permutations

Learn about combinations


Learning Objectives

Explore generalized permutations and combinations

Learn about binomial coefficients and explore the algorithm to compute them

Discover the algorithms to generate permutations and combinations

Become familiar with discrete probability


Basic Counting Principles





There are three boxes containing books. The first box contains 15 mathematics books by different authors, the second box contains 12 chemistry books by different authors, and the third box contains 10 computer science books by different authors.

A student wants to take a book from one of the three boxes. In how many ways can the student do this?



Suppose tasks T1, T2, and T3 are as follows:

T1 : Choose a mathematics book.

T2 : Choose a chemistry book.

T3 : Choose a computer science book.

Then tasks T1, T2, and T3 can be done in 15, 12, and 10 ways, respectively.

All of these tasks are independent of each other. Hence, the number of ways to do one of these tasks is 15 + 12 + 10 = 37.




Basic Counting Principles Morgan is a lead actor in a new movie. She needs to shoot a scene in

the morning in studio A and an afternoon scene in studio C. She looks at the map and finds that there is no direct route from studio A to studio C. Studio B is located between studios A and C. Morgan’s friends Brad and Jennifer are shooting a movie in studio B. There are three roads, say A1, A2, and A3, from studio A to studio B and four roads, say B1, B2, B3, and B4, from studio B to studio C. In how many ways can Morgan go from studio A to studio C and have lunch with Brad and Jennifer at Studio B?



There are 3 ways to go from studio A to studio B and 4 ways to go from studio B to studio C.

The number of ways to go from studio A to studio C via studio B is 3 * 4 = 12.





Consider two finite sets, X1 and X2. Then

This is called the inclusion-exclusion principle for two finite sets.

Consider three finite sets, A, B, and C. Then

This is called the inclusion-exclusion principle for three finite sets.




Pigeonhole Principle

The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.







Permutations


Permutations


Combinations


Combinations


Generalized Permutations and Combinations


Generalized Permutations and Combinations


Binomial Coefficients

The expression x +y is a binomial expression as it is the sum of two terms.

The expression (x +y)n is called a binomial expression of order n.







Pascal’s Triangle

The number C(n, r) can be obtained by constructing a triangular array.

The row 0, i.e., the first row of the triangle, contains the single entry 1. The row 1, i.e., the second row, contains a pair of entries each equal to 1.

Calculate the nth row of the triangle from the preceding row by the following rules:






ALGORITHM 7.1: Determine the factorial of a nonnegative integer. Input: n—a positive integer

Output: n!

1. function factorial(n)

2. begin

3. fact := 1;

4. for i := 2 to n do

5. fact := fact * i;

6. return fact;

7. end



The technique known as divide and conquer can be used to compute C(n, r ).

In the divide-and-conquer technique, a problem is divided into a fixed number, say k, of smaller problems of the same kind.

Typically, k = 2. Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained.

The solutions of the smaller problems are then put together to obtain the solution of the original problem.




Binomial Coefficients ALGORITHM 7.3: Determine C(n, r) using dynamic

programming. Input: n, r , n > 0, r > 0, r ≤ n

Output: C(n, r)

1. function combDynamicProg(n,r)

2. begin

3. for i := 0 to n do

4. for j := 0 to min(i,r) do

5. if j = 0 or j = i then

6. C[i,j] := 1;

7. else

8. C[i,j] := C[i-1, j-1] + C[i-1, j];

9. return C[n, r];

10. end


Generating Permutations and Combinations














Discrete Probability

Definition 7.8.1

A probabilistic experiment, or random experiment, or simply an experiment, is the process by which an observation is made.

In probability theory, any action or process that leads to an observation is referred to as an experiment.

Examples include:

Tossing a pair of fair coins.

Throwing a balanced die.

Counting cars that drive past a toll booth.



Definition 7.8.3

The sample space associated with a probabilistic experiment is the set consisting of all possible outcomes of the experiment and is denoted by S.

The elements of the sample space are referred to as sample points.

A discrete sample space is one that contains either a finite or a countable number of distinct sample points.



Definition 7.8.6

An event in a discrete sample space S is a collection of sample points, i.e., any subset of S. In other words, an event is a set consisting of possible outcomes of the experiment.

Definition 7.8.7

A simple event is an event that cannot be decomposed. Each simple event corresponds to one and only one sample point. Any event that can be decomposed into more than one simple event is called a compound event.



Definition 7.8.8

Let A be an event connected with a probabilistic experiment E and let S be the sample space of E. The event B of nonoccurrence of A is called the complementary event of A. This means that the subset B is the complement A’ of A in S.

In an experiment, two or more events are said to be equally likely if, after taking into consideration all relevant evidences, none can be expected in reference to another.





Axiomatic Approach

Analyzing the concept of equally likely probability, we see that three conditions must hold.

1. The probability of occurrence of any event must be greater than or equal to 0.

2. The probability of the whole sample space must be 1.

3. If two events are mutually exclusive, the probability of their union is the sum of their respective probabilities.

These three fundamental concepts form the basis of the definition of probability.









Conditional Probability

Consider the throw of two distinct balanced dice. To find the probability of getting a sum of 7, when it is given that the digit in the first die is greater than that in the second.

In the probabilistic experiment of throwing two dice the sample space S consists of 6 * 6 = 36 outcomes.

Assume that each of these outcomes is equally likely. Let A be the event: The sum of the digits of the two dice is 7, and let B be the event: The digit in the first die is greater than the second.



Conditional Probability

A : {(6, 1), (5 , 2), (4, 3), (3, 4), (2, 5), (1, 6)}

B : {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (5 , 1), (5 , 2), (5 , 3),(5 , 4), (4, 1), (4, 2), (4, 3), (3, 1), (3, 2), (2, 1)}.

Let C be the event: The sum of the digits in the two dice is 7 but the digit in the first die is greater than the second. Then C : {(6, 1), (5 , 2), (4, 3)} = A ∩ B.

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Discrete Mathematical Structures (Counting Principles)