Discrete Functions
Permutations and Combinations
The Permutation Function
The permutation function is defined as:
P(n, k) =
Examples:
| P(6, 3) | = |
 =  =  = 6(5)(4) = 120. |
|
| P(9, 2) | = |
 =  =  = 9(8) = 72. |
|
| P(7, 1) | = |
 =  = 7 |
|
| P(10, 10) | = |
 =  =  = 10! = 3628800. |
The permutation function yields the number of ways that
n
distinct items can
be arranged in
k
spots. For example,
P(7, 3) = 
= 210
. We can
see that this yields the number of ways 7 items can be arranged in 3 spots --
there are 7 possibilities for the first spot, 6 for the second, and 5 for the
third, for a total of 7(6)(5):
P(7, 3) == 7(6)(5) .
Example: The coach of a basketball team is picking among 11 players for
the 5 different positions in his starting lineup. How many different lineups
can he pick?
P(11, 5) ==
= 55440 different lineups.
The Combination Function
The combination function is defined as:
C(n, k) =
Examples:
| C(6, 3) | = |
 =  = 20. |
|
| C(9, 2) | = |
 =  = 36. |
|
| C(7, 1) | = |
 =  = 7. |
|
| C(10, 10) | = |
 =  = 1. |
The combination function yields the number of ways n distinct items can be chosen for k spots, when the order in which they are chosen does not matter--that is, choosing ABCDE is equivalent to choosing BAEDC. In other words, we use the combination function when all spots are equivalent.
Example: If Jim has 12 shirts, and needs to pack 7 for vacation, how
many different combinations of shirts can he pack?
C(12, 7) == 792 different combinations of shirts.
