MAT 2720 Discrete Mathematics

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MAT 2720 Discrete Mathematics. Section 6.1 Basic Counting Principles. http://myhome.spu.edu/lauw. General Goals. Develop counting techniques. Set up a framework for solving counting problems. The key is not (just) the correct answers.

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### MAT 2720Discrete Mathematics

Section 6.1

Basic Counting Principles

http://myhome.spu.edu/lauw

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
• Inclusion-Exclusion Principle
Example 1

LLL-DDD

# of possible plates = ?

Analysis

LLL-DDD

# of possible plates = ?

Procedure:

Step 1: Step 4:

Step 2: Step 5:

Step 3: Step 6:

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:

:

• Number of possible element that can be selected fromX1or X2or …or Xkis
• 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

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 2720Discrete Mathematics

Section 6.2

Permutations and Combinations Part I

http://myhome.spu.edu/lauw

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?

Example 2

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