MAT 2720Discrete 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

# 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

1Step

1 n ways

Step k

kn ways

3Step

3 n ways

2Step

2 n ways

Number of possible ways to complete the procedure is

1 2 kn n n

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 = ?

EE Department

37 Professors

83 Students

Analysis

Pick a person to joint a university committee.

# of possible ways = ?

The 2 sets: :

EE Department

37 Professors

83 Students

Addition Principle

Number of possible element that can be selected from X1 or X2 or …or Xk is

OR

1X

1 n elements

2X

2 n elements

3X

3 n elements

kX

kn elements

1 2 kn n n

1 2 1 2k kX X X n n n

Example 4

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

Committee

chairperson

secretary

treasurer

A,B,C,D,E,F

Example 4 (a)

In how many ways can this be done?

Committee

chairperson

secretary

treasurer

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?

Committee

chairperson

secretary

treasurer

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?

Committee

chairperson

secretary

treasurer

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?

Committee

chairperson

secretary

treasurer

A,B,C,D,E,F

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

| and X Y x x X x Y

X Y

X Y

X Y52

413

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

| and X Y x x X x Y

X Y

1,2,3 , 3, 4,5

3

X Y

X Y

X Y52

413

Example 5

What is the relationship between

, , , and ?X Y X Y X Y

X Y

1,2,3

3, 4,5

3

1, 2,3,4,5

X X

Y Y

X Y X Y

X Y X Y

X Y

Inclusion-Exclusion Principle

X Y

X YX Y

X Y X Y X Y

Example 4(e)

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

Committee

chairperson

secretary

treasurer

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 2720Discrete 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

CD EF

2nd prize 3rd prize 4th prize

AB

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

1 2, , , nx x x

1 2, , , nx x x

1st 2nd 3rd r - th

3x1x nx

2x

r-permutations

A r-permutation of n distinct objects

is an ordering of an r-element subset of

The number of all possible ordering:

1 2, , , nx x x

1 2, , , nx x x

1st

3x1x nx

2x

2nd 3rd r - th

( , )P n r

Example 1

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

1st prize

CD EF

2nd prize 3rd prize 4th prize

AB

(6,4)P

Theorem

( , ) ( 1) ( 2) ( 1)

!

( )!

P n r n n n n r

n

n r

1st 2nd 3rd r - th

3x1x nx

2x

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?