Discrete Functions

Permutations and Combinations

Summary 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.