Combinations
Suppose six girls from a group of nine are chosen to start the championship
volleyball game. In this case, the order in which the girls are chosen is not
important. Such a selection is called a combination. For example, in the
lottery the winning numbers are “2, 6, 12, and 14”. You don’t need to have
your numbers in the exact same order to win. You could have “12, 14, 6, 2”
and still win, or “6, 2, 14, 12” and still win or …..
Example: How many permutations and combinations are there if you select two digits from
the numbers 1, 2 and 3?
12
13
23
21
31
32
There are 6 permutations and 3 combinations.
The number of combinations of n distinct objects taken r at a
time is defined as follows.
n
!
=
C
(
n
,
r
)
−
(
n
r
)!
r
!
NO REPETITIONS ARE ALLOWED!
Check our answers with the formulas:
! 3
! 3
=
=
C
, 3 (
) 2
P
, 3 (
) 2
−
−
3 (
2
)!
! 2
3 (
2
)!
! 3
! 3
=
=
! 1
! 2
! 1
=
=
6
3
Example 1: A baseball team with 12 players is allowed to send four players to a
weekend batting clinic. In how many ways can the group be chosen?
(This would be a combination question because the order you pick the
groups is not important, just as long as you have four players)
12
!
=
C
(
12
,
) 4
−
(
12
4
)!
! 4
12
!
=
! 8
! 4
=
495