3.5Sample Spaces Having Equally Likely Outcomes 65
relevant (since the first one selected can be any one of then, and the second selected any
one of the remainingn−1, etc.), and since each group ofritems will be countedr! times
in this count, it follows that the number of different groups ofritems that could be formed
from a set ofnitems is
n(n−1)···(n−r+1)
r!=n!
(n−r)!r!NOTATION AND TERMINOLOGY
We define
(n
r)
, forr≤n,by
(
n
r)
=r!
(n−r)!r!and call
(n
r)
the number ofcombinationsofnobjects takenrat a time.
Thus(n
r)
represents the number of different groups of sizerthat can be selected from a
set of sizenwhen the order of selection is not considered relevant. For example, there are
(
8
2)
=8 · 7
2 · 1= 28different groups of size 2 that can be chosen from a set of 8 people, and
(
10
2)
=10 · 9
2 · 1= 45different groups of size 2 that can be chosen from a set of 10 people. Also, since 0!=1,
note that
(
n
0
)
=(
n
n)
= 1EXAMPLE 3.5d A committee of size 5 is to be selected from a group of 6 men and 9 women.
If the selection is made randomly, what is the probability that the committee consists of
3 men and 2 women?
SOLUTION Let us assume that “randomly selected” means that each of the
( 15
5)
possiblecombinations is equally likely to be selected. Hence, since there are
( 6
3)
possible choicesof 3 men and
( 9
2)
possible choices of 2 women, it follows that the desired probability is
given by
(
6
3
)(
9
2)(
15
5) =240
1001■