Permutations & Combinations Calculator

Permutations without repetition — P(n, k) = n! / (n−k)!. Choose k of n items where order matters and no item repeats. Example: how many ways to seat 3 of 10 people in a 1st/2nd/3rd-place podium.

Combinations without repetition — C(n, k) = n! / (k!(n−k)!). Order doesn't matter, no repeats. Example: 5-card poker hands from a 52-card deck.

Permutations with repetition — nᵏ. k slots, each independently filled from n choices. Example: 4-digit PIN from 10 digits = 10⁴ = 10000.

Combinations with repetition (multisets) — C(n+k−1, k). k items chosen from n types, repeats allowed, order doesn't matter. Example: scooping 3 ice-cream balls from 5 flavours.

A 'pick 6 from 49' lottery: how many possible tickets? Order doesn't matter and no number repeats — combinations without repetition.

C(49, 6) = 49! / (6! · 43!) = 13 983 816. One ticket has odds 1 in 13.98 million.

By contrast, a 6-digit number with repeats allowed = 10⁶ = 1 000 000 — much better odds, but far smaller prize pools because the count is so much smaller.

8-character password from 62 alphanumeric characters with repeats: 62⁸ ≈ 2.18 × 10¹⁴. Why a brute-forcer needs an offline hash to crack reasonable passwords.

Poker hands: total = C(52, 5) = 2 598 960. Royal flushes = 4. P(royal) = 4 / 2598960 ≈ 1 in 650 000.

Same machinery — 'choose k of n' — applied at very different scales.

C(n, k) = C(n, n−k). Choosing the 'in' group is the same problem as choosing the 'out' group.

P(n, k) = k! · C(n, k). Permutations equal combinations multiplied by the orderings of the chosen k items.

n! grows ferociously: 10! ≈ 3.6 million, 20! ≈ 2.4 × 10¹⁸, 70! overflows IEEE 754 double precision. This calculator caps at 170! (≈ 1.7 × 10³⁰⁶) — anything beyond returns Infinity.

Treating ordered problems as unordered. 'How many possible podiums?' is P, not C.

Forgetting repetition. 'How many 7-digit phone numbers?' uses repetition (10⁷); ranking 7 distinct candidates doesn't (7!).

Multinomial coefficients — arranging n items where some repeat (like the letters in MISSISSIPPI) need n! / (n₁! · n₂! · …). Not covered by the four rules above.

In computer science, combinatorics drives algorithm analysis. Sorting n items has n! possible initial orderings, which is why comparison sorts can't beat O(n log n) in the worst case. Password entropy is log₂(nᵏ) — the number of bits of security in a k-character password drawn from n symbols.

In genetics, combinations count possible genotypes. With two alleles at a single locus, there are C(2+2−1, 2) = 3 genotypes (AA, Aa, aa). Across 20,000+ genes, the combinatorial space of human variation is astronomical — which is why twin studies focus on comparing identical versus fraternal pairs to isolate genetic from environmental effects.