Thinking Mathematically

Counting Methods and Probability11.1 The Fundamental Counting Principle

The Fundamental Counting Principle

If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is M N.

Exercise 11.1 #3A popular brand of pen is available in three colors (red, green, or

blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

The Fundamental Counting Principle

The number of ways a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

Example: The Fundamental Counting Principle

Exercise 11.1 #9

A restaurant offers the following lunch menu

Main Course Vegetables Beverages Desserts

Ham Potatoes Coffee Cake

Chicken Peas Tea Pie

Fish Green beans Milk Ice cream

Beef Soda

If one item is selected from each of the four groups, in how many different ways can a meal be ordered?

Example Telephone Numbers in the United States

Exercise Set 11.1 #17

In the original (1945) plan for area codes, the first digit could be 2 through 9, the second 0 or 1, and the third any digit any number except 0. How many different area codes are possible under this plan?

Thinking Mathematically

Counting Methods and Probability11.2 Permutations

Permutations

• No item is used more than once.

• The order of arrangement makes a difference.

Examples: Permutations

Exercise 11.2 #7, 11You need to arrange nine of your favorite books

along a shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference?

How many ways can five movies be ordered into two hour time slots between 6PM and 4AM. Two of the movies have G ratings and are to be shown in the first two slots. One is rated NC-17 and is to be shown in the last time slot.

Factorial Notation

If n is a positive integer, the notation n! is the product of all positive integers from n down through 1.

n! = n(n-1)(n-2)…(3)(2)(1)

0!, by definition is 1.

0!=1

Exercise 11.2 #15

!25

!29

Permutations of n Things Taken r at a Time

The number of permutations possible if r items are taken from n items:

Exercise 11.2 #35, 45

Evaluate 8P5

In a race in which six automobiles are entered and there are no ties, in how many ways

can the first three finishers come in ?

Thinking Mathematically

Counting Methods and Probability11.3 Combinations

A combination of items occurs when:

• The items are selected from the same group.

• No item is used more than once.• The order of the items makes no

difference.• How is this different from a

permutation?

Permutations or Combinations?

Exercise Set 11.3 #1, 3• A medical researcher needs 6 people to test the

effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

• How many different four-letter passwords can be formed from the letters A, B, C, D, E, F, and G if no repetition of letters is allowed?

!)!(

!

rrn

nCrn

Combinations of n Things Taken r at a Time

The number of possible combinations if r items are taken from n items is

Exercise 11.3 #9Evaluate 11C4

Thinking Mathematically

Chapter 11

Counting Methods and Probability