Page 114 - Probability Demystified

CHAPTER 6 The Counting Rules 103

Suppose two letters are selected from the four letters, A, B, C, and D.
The different permutations are shown on the left and the different combina-
tions are shown on the right.
PERMUTATIONS COMBINATIONS
AB BA CA DA AB BC
AC BC CB DB AC BD
AD BD CD DC AD CD
Notice that in a permutation AB differs from BA, but in a combination
AB is the same as BA. The combination rule is used to find the number of
ways to select objects without regard to order.
The number of ways of selecting r objects from n objects without regard to
order is
n!
C ¼
n r
ðn rÞ!r!
Note: The symbol n C r is used for combinations; however, some books use
n n
other symbols. Two of the most commonly used symbols are C or ð Þ:
r
r
EXAMPLE: In how many ways can 2 objects be selected from 6 objects
without regard to order?

SOLUTION:

Let n ¼ 6 and r ¼ 2,
6! 6! 6 5 4!
C ¼ ¼ ¼ ¼ 15
6 2
ð6 2Þ!2! 4!2! 4! 2 1

EXAMPLE: A salesperson has to visit 10 stores in a large city. She decides to
visit 6 stores on the first day. In how many different ways can she select the
6 stores? The order is not important.

SOLUTION:
Let n ¼ 10 and r ¼ 6; then

10! 10! 10 9 8 7 6!
10 C ¼ ¼ ¼ ¼ 210
6
ð10 6Þ!6! 4!6! 4 3 2 1 6!
She can select the 6 stores in 210 ways.