$Permutations and Combinations$

Permutations and Combinations

Combinatorics calculator: the number of permutations, arrangements (nPr) and combinations (nCr), with and without repetition, with formula and solution.

What to find:

n and k are whole numbers up to 1000. For permutations with repetition enter the counts of identical elements separated by spaces: for the word "MAMA" — 2 2.

Test yourself

The combinatorics calculator finds the number of permutations, arrangements (k-permutations) and combinations — without and with repetition. The result is exact even with hundreds of digits, and the solution shows the formula and the meaning of the answer.

Main formulas

What we count	Order matters?	Formula
Permutations of n	yes, all elements	P_n = n!
Arrangements of n by k	yes	A_n^k = n! / (n − k)!
Combinations of n by k	no	C_n^k = n! / (k!(n − k)!)
Arrangements with repetition	yes	n^k
Combinations with repetition	no	C_n+k−1^k
Multiset permutations	yes	n! / (n₁! · n₂! · … · n_k!)

How to choose the formula

Does order matter? A podium (1st, 2nd, 3rd place) — yes, arrangements: A₁₀³ = 720. A team of 3 — no, combinations: C₁₀³ = 120.
Can elements repeat? A three-digit code from digits 0–9 — yes: 10³ = 1000.
Are all elements rearranged? Arranging 5 books on a shelf: P₅ = 120. Anagrams of "MAMA": 4!/(2! · 2!) = 6.

Example. How many 5-card hands can be dealt from a 52-card deck?
C₅₂⁵ = 52 · 51 · 50 · 49 · 48 / 120 = 2,598,960.