Algebra 1 Common Core Student Edition, Grade 8-9

How do you use
the Multiplication
Count ing Princip le t o
find the number of
permutations?
Multiply the number
of ways to make each
selection.

Fi n d i n g Pe r m u t a t i o n s

Baseb al l How many different batting orders can you have with 9 players?

There are 9 choices for the first batter, 8 for the second, and so on.

9'8,7*6,5*4*3*2*1= 362,880 Use a calculator.

There are 362,880 possible batting orders.

Go t I t? 2. A swimming pool has 8 lanes. In how many ways can 8 swimmers be

assigned lanes for a race?

A short way to write the product in Problem 2 is 9!, read "nine factorial.” For any positive

integer n, the expression n factorial is written as n\ and is the product of the integers

from n down to 1. The value of 0! is defined to be 1.

You can use factorials to write a formula for the number of permutations of n objects

arranged r at a time.

Key Concept Per m u t at i o n N o t at i o n

The expression nPr represents the number of permutations of n objects arranged

r at a time.

p — ul_

n r (n — r)\

r i r, 8! _8!_8'7,6*5,4'3'2,l_rc

Ex a m p l e 8P 2 (8 _ 2)i 6! 6 • 5 • 4 • 3 • 2 • l 56

Pr o b l em 3 Using Perm utation Notation

How can you t hink

about t his problem in

a d i f f er en t w ay?

You can use the

Multiplication Counting

Principle. There are

7 choices for the first

song, 6 for the second,

5 for the third, 4 for the

fourth, and 3 for the fifth.

Music A band has 7 new songs and wants to put 5 of them on a demo CD. How many

arrangements of 5 songs are possible?

To find the number of possible arrangements, find the value of 7P5.

Method 1 Use the formula for permutations.

7! 7! 7 * 6 * 5 * 4 * 3 * 2 * 1

7P5 (7-5)! 2!

= 2520

2 • 1

Method 2 Use a graphing calculator.

Press

7P5 = 2520

There are 2520 possible arrangements of 5 songs.

Write using

factorials.

Simplify.

764 Ch ap t er 12 Data Analysis and Probability