MAT 2720 Discrete Mathematics Section 6.1 Basic Counting Principles

General Goals Develop counting techniques. Set up a framework for solving counting problems. The key is not (just) the correct answers. The key is to explain to your audiences how to get to the correct answers (communications).

Goals Basics of Counting Multiplication Principle Addition Principle Inclusion-Exclusion Principle

Example 1 License Plate # of possible plates = ? LLL-DDD

Analysis License Plate # of possible plates = ? Procedure: Step 1:Step 4: Step 2:Step 5: Step 3:Step 6: LLL-DDD

Multiplication Principle Suppose a procedure can be constructed by a series of steps Number of possible ways to complete the procedure is

Example 2(a) Form a string of length 4 from the letters A, B, C, D, E without repetitions. How many possible strings?

Example 2(b) Form a string of length 4 from the letters A, B, C, D, E without repetitions. How many possible strings begin with B ?

Example 3 Pick a person to joint a university committee. # of possible ways = ?

Analysis Pick a person to joint a university committee. # of possible ways = ? The 2 sets: :

Addition Principle Number of possible element that can be selected from X 1 or X 2 or …or X k is OR

Example 4 A 6-person committee composed of A, B, C, D, E, and F is to select a chairperson, secretary, and treasurer.

Example 4 (a) In how many ways can this be done?

Example 4 (b) In how many ways can this be done if either A or B must be chairperson?

Example 4 (c) In how many ways can this be done if E must hold one of the offices?

Example 4 (d) In how many ways can this be done if both A and D must hold office?

Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set

Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set

Example 5 What is the relationship between

Inclusion-Exclusion Principle

Example 4(e) How many selections are there in which either A or D or both are officers?.

Remarks on Presentations Some explanations in words are required. In particular, when using the Multiplication Principle, use the “steps” to explain your calculations A conceptual diagram may be helpful.

MAT 2720 Discrete Mathematics Section 6.2 Permutations and Combinations Part I

Goals Permutations and Combinations Definitions Formulas Binomial Coefficients

Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible? Step 1: Step 2: Step 3: Step 4:

r-permutations A r-permutation of n distinct objects is an ordering of an r-element subset of

r-permutations A r-permutation of n distinct objects is an ordering of an r-element subset of The number of all possible ordering:

Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible?

Theorem

Example persons enter into a contest. How many possible ways to select the 1 st, 2 nd, and 3 rd prize winner?

Example 3(a) How many 3-permutations of the letters A, B, C, D, E, and F are possible?

Example 3(b) How many permutations of the letters A, B, C, D, E, and F are possible. Note that, “permutations” means “6- permutations”.

Example 3(c) How many permutations of the letters A, B, C, D, E, and F contains the substring DEF ?

Example 3(d) How many permutations of the letters A, B, C, D, E, and F contains the letters D, E, and F together in any order?