Permutation Calculator

Permutation Calculator: solve permutation problems step-by-step. Clear formula, worked example, and FAQs included.

A combinations and permutations calculator counts the number of ways to select or arrange items from a set. Permutations count ordered arrangements; combinations count unordered selections. These calculations underpin probability theory, statistics, cryptography, and combinatorics. Factorials (n!) are the foundation: n! = n × (n−1) × ... × 1. Percentage Calculator and Ratio Calculator provide related tools.

For combinations C(n,r): choose r items from n without regard to order. Enter n (total items) and r (items chosen).
For permutations P(n,r): arrange r items from n where order matters.
For factorial n!: enter n to compute n × (n−1) × ... × 1.

Combinations and permutations

Combinations: C(n,r) = n! ÷ (r! × (n−r)!)

Permutations: P(n,r) = n! ÷ (n−r)!

Factorial: n! = n × (n−1) × ... × 2 × 1; 0! = 1

Example: Choosing 3 from 10. C(10,3) = 120 (unordered). P(10,3) = 720 (ordered).

Combinations vs permutations

When to use each

Use combinations when order doesn't matter: selecting a committee, choosing lottery numbers, picking a subset. Use permutations when order matters: rankings, passwords, sequences. The ratio P(n,r)/C(n,r) = r! — permutations always exceed combinations by a factor of r! for the same n and r.

Math tips and best practices

C(n,r) = C(n, n−r) — choosing 3 from 10 gives the same count as choosing 7 from 10.
For large n, use Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ.
Pascal's triangle generates combinations: each entry is C(row, position).

The number of possible 6/49 lottery combinations is C(49,6) = 13,983,816 — approximately 1 in 14 million odds.
A standard 52-card deck has C(52,5) = 2,598,960 possible 5-card hands.

Common mistakes to avoid

Using permutations when order doesn't matter — vastly overcounts unordered selections.
Forgetting that 0! = 1 by definition — this is essential for C(n,0) = 1 and C(n,n) = 1 to work.

Combinatorial calculations are exact for integer inputs. For very large factorials, numerical overflow may occur — use logarithmic methods for n > 170. This calculator is for educational purposes.