Page 111 - Probability Demystified

100 CHAPTER 6 The Counting Rules

EXAMPLE: In how many different ways can 6 people be arranged in a row
for a photograph?

SOLUTION:

This is a permutation of 6 objects. Hence 6! ¼ 6 5 4 3 2 1 ¼ 720 ways.

In the previous example, all the objects were used; however, in many
situations only some of the objects are used. In this case, the permutation rule
can be used.
The arrangement of n objects in a specific order using r objects at a time is
called a permutation of n objects taking r objects at a time. It is written as n P r
and the formula is
n!
n P ¼
r
ðn rÞ!

EXAMPLE: In how many different ways can 3 people be arranged in a row
for a photograph if they are selected from a group of 5 people?

SOLUTION:
Since 3 people are being selected from 5 people and arranged in a specific
order, n ¼ 5, r ¼ 3. Hence, there are
5! 5! 5 4 3 2!
P ¼ ¼ ¼ ¼ 5 4 3 ¼ 60 ways
5 3
ð5 3Þ! 2! 2!

EXAMPLE: In how many different ways can a chairperson and secretary be
selected from a committee of 9 people?

SOLUTION:

In this case, n ¼ 9 and r ¼ 2. Hence, there are 9 P 2 ways of selecting two people
to fill the two positions.
9! 9! 9 8 7!
2
9 P ¼ ¼ ¼ ¼ 72 ways
ð9 2Þ! 7! 7!