MAT 2720 Discrete Mathematics

Section 6.1 Basic Counting Principles

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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

LLL-DDD

# of possible plates = ?

Analysis

License Plate

LLL-DDD

# of possible plates = ?

Procedure: Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

Multiplication Principle

Suppose a procedure by a series of steps can be constructed

Step

1

Step

2

Step

3

n ways

2

k

Number of possible ways to complete the procedure is 1    2

n k

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.

37

Professors

83

Students

# of possible ways = ?

Analysis

Pick a person to joint a university committee.

37

Professors

83

Students

# of possible ways = ?

The 2 sets: :

Addition Principle

X

1

X

2

X

3

n elements

1

n elements

2

n elements

3

k X k

 Number of possible element that can be selected from

X 1

or

X 2

or

…or X k

is

n

1

n k

 OR

X

1 

X

2  

X k

1

n

2

n k

Example 4

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

chairperson secretary treasurer

Committee

A,B,C,D,E,F

Example 4 (a)

In how many ways can this be done?

chairperson secretary treasurer

Committee

A,B,C,D,E,F

Example 4 (b)

In how many ways can this be done if either A or B must be chairperson?

chairperson secretary treasurer

Committee

A,B,C,D,E,F

Example 4 (c)

In how many ways can this be done if E must hold one of the offices?

chairperson secretary treasurer

Committee

A,B,C,D,E,F

Example 4 (d)

In how many ways can this be done if both A and D must hold office?

chairperson secretary treasurer

Committee

A,B,C,D,E,F

Recall: Intersection of Sets (1.1)

X

The intersection the set of X and Y is defined as  

X

and  

X



Y X Y

Recall: Intersection of Sets (1.1)

X

The intersection the set of X and Y is defined as  

X

and  

X



Y X

  

Y

  3, 4, 5 

X

1

X

2 3 4

Y

5

Example 5

What is the relationship between ,

X



Y

, and

X



Y

?

1

X

2

X



Y

3

X



Y

4

Y

5

Y X X X

       

X

 

X X Y

 

Inclusion-Exclusion Principle

X X



Y



X



Y X X



Y X



Y Y

Example 4(e)

How many selections are there in which either A or D or both are officers?.

chairperson secretary treasurer

Committee

A,B,C,D,E,F

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

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Goals

 Permutations and Combinations • Definitions • • Formulas Binomial Coefficients

Example 1

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

1st prize D 2nd prize F 3rd prize C 4th prize E B A Step 1: Step 2: Step 3: Step 4:

r-permutations

A r-permutation of n distinct objects

x x

1 , 2 , is an ordering of an r-element subset of 

x x

1 , 2 , ,

x n

 ,

x n

1st x 1 2nd x 2 3rd x 3 r-th x

n

r-permutations

A r-permutation of n distinct objects

x x

1 , 2 , is an ordering of an r-element subset of 

x x

1 , 2 , ,

x n

 The number of all possible ordering: ,

x n

1st x 1 2nd x 2 3rd x 3 r-th x

n

Example 1

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

P

(6, 4)  1st prize D 2nd prize F 3rd prize C 4th prize E B A

Theorem

 (

n

!

)!

1) (

n

2) ( 1) 1st x 1 2nd x 2 3rd x 3 r-th x

n

Example 2

100 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?