What Is a Permutations Calculator?

A permutations calculator computes P(n, r) — the number of ordered arrangements of r items chosen from n distinct items. Unlike combinations (where order doesn't matter), permutations count each different ordering as a separate outcome. This calculator shows both the permutation count and the corresponding combination count, plus a bar chart showing how permutations grow as r increases from 0 to the entered value.

Pre-filled with P(10, 3) = 720, the calculator shows that there are 720 different ways to arrange 3 items chosen from 10 when the order matters — for example, awarding gold, silver, and bronze medals to 3 of 10 athletes.

The Permutations Formula

P(n, r) = n! / (n − r)!

For P(10, 3): P = 10! / 7! = (10 × 9 × 8 × 7!) / 7! = 10 × 9 × 8 = 720. Notice you can cancel the (n−r)! factorial and just multiply n × (n−1) × (n−2) × … for r terms. This shortcut avoids computing large factorials.

Permutations vs Combinations

The key relationship: P(n, r) = C(n, r) × r!. For every unordered combination of r items, there are r! ways to arrange them. So permutations always exceed combinations (when r > 1). P(10,3) = 720 while C(10,3) = 120. The ratio 720/120 = 6 = 3! = the number of orderings of 3 items.

Use permutations when order matters: Race placements (1st ≠ 2nd), PIN combinations (1234 ≠ 4321), phone number arrangements, seating arrangements, scheduling ordered lists.

Use combinations when order doesn't matter: Choosing committee members, card hands, lottery picks, pizza topping selections.

Counting with Repetition

Standard permutations assume no repetition — each item can only appear once. If repetition is allowed (like a 4-digit PIN where digits can repeat), the count is simply n^r. For a 4-digit PIN from digits 0–9: 10⁴ = 10,000 possible codes. The permutations formula P(10,4) = 5,040 gives only the count where each digit appears at most once.

Practical Examples

Olympics medals: 8 swimmers compete; how many ways can gold/silver/bronze be awarded? P(8,3) = 8!/5! = 336 possible podium results.

Seating arrangement: 6 people seated in a row: P(6,6) = 6! = 720 arrangements. At a round table (circular permutations): (6−1)! = 120, since rotations of the same arrangement are identical.

Password strength: An 8-character password from 26 lowercase letters without repetition: P(26,8) = 26!/(26-8)! = 62,990,928,000 — about 63 billion combinations. Pair with the combinations calculator for complete counting analysis.

Tips for Using This Calculator

Enter n and r as non-negative integers with r ≤ n. Maximum n = 20 due to integer overflow. The growth chart shows how rapidly permutations increase as r grows — a powerful visual for understanding combinatorial explosion. For larger values, use Stirling's approximation or logarithms to estimate factorial-based formulas.