How to Calculate Permutations nPr

Calculates the number of ways to arrange r items chosen from n, nPr = n × (n−1) × … × (n−r+1), where the order matters.

Explanation

A permutation counts the ways of choosing $r$ items from $n$ and arranging them in a row. The order matters.

_nP_r = n \times (n-1) \times \cdots \times (n-r+1) = \dfrac{n!}{(n-r)!}

It is a product of $r$ factors: $n$ choices for the first place, $n-1$ for the second, and so on down.

Example

Pick 3 people out of 5 and award first, second and third place.

_5P_3 = 5 \times 4 \times 3 = 60

There are 5 candidates for first place, 4 remaining for second and 3 for third, giving $5 \times 4 \times 3 = 60$ arrangements.

The only difference is whether the order counts.

Permutation — "A first, B second" and "B first, A second" are different outcomes
Combination — both simply select A and B, so they are the same outcome

A permutation count is $r!$ times the combination count, because each set of $r$ chosen items can be arranged in $r!$ orders.

_nC_r = \dfrac{_nP_r}{r!}

Use permutations when the arrangement carries meaning — rankings, seating, PIN codes. Use combinations when you are only selecting.