Warm-Up Problem

Can you predict which offers more choices for license plates?

Choice A: a plate with three different letters of the alphabet in any

orderChoice B: a plate with four different

nonzero digits in any order.

By the end of class, you should be able to successfully answer this question.

Example 1

Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box.

A second piece of paper is drawn from the box, and its number is written down. If the two numbers are added together, how many different ways can a total of 12 be obtained?

Example 2

Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box and its number is written down.

A second piece of paper is drawn from the box, and its number is written down.

If the two numbers are added together, how many different ways can a total of 12 be obtained?

Example 3

You have three books that you want to place on a shelf.

How many different ways can you arrange the books on the shelf?

The Counting Principle

Let E1 and E2 be two events. The first event E1 can occur m different ways.

The second event E2 can occur n different ways.

The number of ways that the two events can occur is m times n.

Example 3 again.

You have three books that you want to place on a shelf.

How many different ways can you arrange the books on the shelf?

6 different ways to place 3 books on a shelf.

Example 4

You have five books and you want to place three of them on a shelf.

How many different ways can you arrange three of the five books on the shelf?

60 different ways

This is the number of different arrangements or permutations of 5 things taken 3 at a time.

This can be written as P(5,3).

Remember: Permutations of n Elements taken r at a time is found by

n r

n!P n, r P

n r !

Key Words: arrangements, schedule, order

Example 5

Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? Assume there are no ties.

Using the Counting Principle: 8(7)(6)=336.

8 38!

P 3368 3 !

Using Permutations:

Example 6Your teacher gives you a list of three books. You must choose two of the books on the list to read.

How many different pairings can you choose from?

Combinations

When you choose r elements from a large set of n elements and ORDER DOES NOT MATTER,

You are finding the number of combinations of n elements taken r at a time.

n r

n n!C n, r C

r r ! n r !

Key Words: group, committee, selection, sample

Example 7

How many different committees of 3 people can be chosen from a group of 8 people?

Since order does not matter, use combinations

8 8!3 3! 8 3 !

56 committees

Warm-Up Problem

Can you predict which offers more choices for license plates?

Choice A: a plate with three different letters of the alphabet in any

orderChoice B: a plate with four different

nonzero digits in any order.

Alphabet – 15,600 Numbers - 3024