Permutations

Calculator Setup

P(10, 3)

n (total items)

r (chosen)

Choose r items from n distinct items in order. r must be ≤ n.

Examples

n (compute n!)

Computes n! = n × (n−1) × … × 2 × 1. Defined as 0! = 1.

Examples

Result

Enter values above and press Calculate to see the result, falling factorial expansion, and a real-world interpretation.

P(n,r)

—

Falling Factorial Expansion

—

Factorial Form

—

🌐

Permutation Formula

P(n,r) = n! / (n−r)!

Order matters — ABC and BAC are different permutations. When you fill the first slot you have n choices, then n−1 remain for the second, and so on.

The falling factorial form makes this concrete:

P(n,r) = n × (n−1) × (n−2) × … × (n−r+1)

Special cases: P(n,0) = 1 (one way to choose nothing), P(n,n) = n! (arrange all items).

For each slot, one fewer choice remains — that's the core idea behind the falling factorial.

Permutation vs Combination

P(n,r) = r! × C(n,r)

Permutations count ordered arrangements — the order ABC ≠ BAC ≠ CAB.

Combinations count unordered subsets — {A,B,C} is the same group regardless of listing order.

Since each combination of r items can be arranged in r! ways, multiplying C(n,r) by r! recovers P(n,r).

Order matters → use permutation P(n,r)
Order doesn't matter → use combination C(n,r)
Passwords, rankings, seating → permutations
Committees, hands of cards → combinations

Permutations and counting still tricky?