Permutation Calculator

Permutations with repetition

The nPr formula assumes each item is used at most once. When items may repeat, as with the digits of a PIN, each of the r positions has n choices, so the count is n^r. A 4-digit PIN made from 10 digits has 10^4 = 10,000 possibilities, while a 4-digit code with no repeated digits has 10P4 = 10 x 9 x 8 x 7 = 5,040.

Arranging letters that include duplicates needs a different adjustment: divide n! by the factorial of each repeat count. The word LEVEL has 5 letters with L twice and E twice, so its distinct arrangements are 5! / (2! x 2!) = 120 / 4 = 30.

Picking the right model first is the main skill. Ask whether order matters, and whether an item can be used again, before choosing a formula.

Permutations: arrangements where order matters

A permutation counts the ordered arrangements of r items chosen from n distinct items: P(n, r) = n! ÷ (n − r)!. Awarding gold, silver and bronze among 10 runners gives P(10, 3) = 10 × 9 × 8 = 720 possible podiums. Arranging all n items is simply n!; five books on a shelf can be arranged in 5! = 120 ways.

Repeated items and repetition

When some items are identical, divide by the factorial of each repeat count. The letters of LEVEL (two Ls, two Es, one V) can be arranged in 5! ÷ (2! × 2!) = 120 ÷ 4 = 30 distinct ways.

If items can be reused, the count is nʳ instead. A 4-digit PIN using digits 0–9 with repetition allowed has 10⁴ = 10,000 possibilities, while a 4-digit code with no repeated digit has P(10, 4) = 5,040.

Choosing the right formula

Order matters and no repeats: P(n, r). Order matters with repeats: nʳ. Order does not matter: combinations, C(n, r) = P(n, r) ÷ r!. Seating problems around a round table use (n − 1)! because rotating everyone gives the same arrangement, so 6 people can sit in 5! = 120 distinct ways.