When the order of items matters, that’s called a Permutation.
When the order of items doesn’t matter, that’s called a Combination. Since we are not allowed to repeat items, we use the following formula:
Number of possible Permutations
=
n^{r}
=
^{}
=
Number of possible Permutations
=
n!(n – r)!
=
!
( – )!
=
Number of possible Combinations
=
(n + r – 1)!
r!(n – 1)!
=
( + – 1)!!( – 1)!
=
Number of possible Combinations
=
n!r!(n – r)!
=
!
!( – )!
=
The Visual Way
A form of the permutation problem that students commonly see is the “committee” problem. For example:
If there are 5 people, Jim, Jane, Bob, Susan, and Ralph, and only 3 of them can be on the new PTA committee, how many different combinations are possible?
In this example, there are 5 people to choose from (so n equals 5), and we need to choose 3 of them (so r equals 3).
Order doesn’t matter: if Jim is on the committee, he’s on the committee whether he’s picked first or last. Repetition isn’t allowed because Susan can’t be on the committee twice (even if she really wants to be!)
So, if we use the “mathy” way from above, we know the formula is:
Number of possible Combinations
=
n!r!(n – r)!
And we input the number 5 for n, and 3 for r, and so we know that there 10 possible combinations. But what does that actually mean?