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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 2720 discrete mathematics

MAT 2720Discrete Mathematics

Section 6.1

Basic Counting Principles

http://myhome.spu.edu/lauw

general goals
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
Goals
  • Basics of Counting
    • Multiplication Principle
    • Addition Principle
    • Inclusion-Exclusion Principle
example 1
Example 1

LLL-DDD

License Plate

# of possible plates = ?

analysis
Analysis

LLL-DDD

License Plate

# of possible plates = ?

Procedure:

Step 1: Step 4:

Step 2: Step 5:

Step 3: Step 6:

multiplication principle
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
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
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
Example 3

Pick a person to joint a university committee.

# of possible ways = ?

analysis1
Analysis

Pick a person to joint a university committee.

# of possible ways = ?

The 2 sets:

:

addition principle
Addition Principle
  • Number of possible element that can be selected fromX1or X2or …or Xkis
  • OR
example 4
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
Example 4 (a)

In how many ways can this be done?

example 4 b
Example 4 (b)

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

example 4 c
Example 4 (c)

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

example 4 d
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
Recall: Intersection of Sets (1.1)

The intersection of X and Y is defined as the set

recall intersection of sets 1 11
Recall: Intersection of Sets (1.1)

The intersection of X and Y is defined as the set

example 5
Example 5

What is the relationship between

example 4 e
Example 4(e)

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

remarks on presentations
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 mathematics1

MAT 2720Discrete Mathematics

Section 6.2

Permutations and Combinations Part I

http://myhome.spu.edu/lauw

goals1
Goals
  • Permutations and Combinations
    • Definitions
    • Formulas
    • Binomial Coefficients
example 11
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
r-permutations

A r-permutation of n distinct objects

is an ordering of an r-element subset of

r permutations1
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 12
Example 1

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

example 2
Example 2

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

example 3 a
Example 3(a)

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

example 3 b
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
Example 3(c)

How many permutations of the letters A, B, C , D, E, and F contains the substring DEF?

example 3 d
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?