Permutations Calculator
Enter the size of your set and how many you arrange, and get the number of ordered arrangements — the count behind race finishes, passwords, and rankings.
Two values, one count
Enter the total number of items (n) and how many you arrange (r); the calculator returns P(n, r) with the formula P(n, r) = n! ÷ (n − r)!.
Order matters
Permutations count arrangements, not selections — the same items in a different order are counted as separate permutations.
What is a permutation?
Arranging a group when order matters
A permutation is a way of arranging r items chosen from a set of n items where the order of the arrangement matters — the sequence (A, B, C) is a different permutation from (C, A, B). The number of such arrangements is written P(n, r), read "n permute r", and is given by the formula P(n, r) = n! ÷ (n − r)!. Permutations answer "how many different ordered arrangements can I form", which is exactly the question behind podium finishes, password counts, and the seedings of a tournament.
Enter your total items and how many you arrange to get the number of permutations instantly.
One formula, built from the set size n and the number arranged r.
P(n, r) = n! ÷ (n − r)!The factorial n! multiplies every whole number from 1 up to n. Dividing n! by (n − r)! cancels every factor from 1 up to n − r, leaving just the r descending factors n × (n − 1) × … × (n − r + 1). In practice this tool multiplies those r factors directly, so even large values of n never overflow through an intermediate n!.
Suppose you want to award gold, silver, and bronze among 10 runners.
Identify n and r
n = 10 total runners, r = 3 placed — order matters, since gold differs from bronze.
Apply the formula
P(10, 3) = 10! ÷ (10 − 3)! = 10! ÷ 7!.
Simplify
10 × 9 × 8 = 720 — there are 720 possible podium orders.
The number you get is the count of distinct ordered arrangements you can form, with every reordering of the same items counted as its own permutation. The first thing to notice is that permutations are always at least as many as combinations: because each group of r items can be ordered in r! different ways, P(n, r) = C(n, r) × r!, so arranging is order-sensitive and the count grows quickly. The edges are tidy: P(n, 0) = 1, since there is exactly one way to arrange nothing, and P(n, n) = n!, the number of ways to put the whole set in order. Read a large result as a measure of how many ordered outcomes are possible — 311,875,200 arrangements of a 5-card ordered draw from a 52-card deck means each specific dealt sequence is rare. The faster-than-linear growth is why even modest sets produce enormous arrangement counts.
The permutations formula is exact, but a couple of points keep it from being misapplied.
Order on, no repeats, r ≤ n
This calculator counts permutations — arrangements where order matters and each item is used at most once. If order does not matter, you want combinations instead; if items may repeat, you need permutations with repetition, a different formula. Both n and r must be whole numbers, and you cannot arrange more items than exist, so r must be no greater than n — otherwise there is no valid arrangement and the calculator returns no result.