Last updated: 2026-05-22

Definition

A permutation $P(n, r)$ counts the number of ways to choose $r$ items from a set of $n$ distinct items and arrange them in a specific order. Because the sequence matters, selecting {A, B} is a different outcome from {B, A}.

The formula is:

Worked example

How many 3-letter codes can be formed from the letters {A, B, C, D, E} without repeating any letter?

Intuitively: 5 choices for the first position, 4 for the second (one letter is gone), 3 for the third — giving $5 \times 4 \times 3 = 60$.

Permutation vs. combination

The distinction comes down to whether order matters.

The relationship between the two: $P(n, r) = C(n, r) \times r!$. Each combination corresponds to $r!$ permutations — all the orderings of the same selected group. Use the Combination Calculator — C(n, r) to compute $C(n, r)$ directly.

Special cases

$r = 0$: $P(n, 0) = 1$ for any $n$. There is exactly one way to arrange zero items.

$r = n$: $P(n, n) = n!$. This is the total number of arrangements of all $n$ items (see the Factorial Calculator for large $n!$ values).

$r > n$: Undefined — you cannot select more items than exist. The calculator reports an error.

Common applications

• Race results: finishing positions in a race (order determines 1st/2nd/3rd)
• Passwords and PINs: sequences where each character position is distinct
• Scheduling: assigning tasks to time slots in sequence
• Seating arrangements: placing people in numbered chairs

Practical notes

Large results. $P(20, 10) = 670{,}442{,}572{,}800$ — over 670 billion. Even within the $n \leq 20$ limit, results can be very large. This calculator returns exact integer values throughout.

With repetition. The formula above assumes without replacement: no item appears twice. If repetition is allowed (e.g., a PIN where the same digit can repeat), the count is simply $n^r$.