Permutations & Combinations
nPr, nCr, n! and the with-repetition counts nʳ and C(n+r−1, r), circular arrangements and Pascal’s row — exact big integers for n, r up to 5 000.
Permutations and combinations with and without repetition, the factorials, circular permutations, the probability of one particular selection, Pascal’s triangle row and digit counts.
Example: 10 items, 3 chosen: 720 ordered, 120 unordered, 1 000 with repetition in order, 220 unordered with repetition; a 5-card hand from 52 is one of 2 598 960 (52! has 68 digits).
Exact to the last digit,
however many.
The definitions, why big integers are used, and what the extra counts mean.
Definitions
nPr = n!/(n − r)! counts ordered selections without repetition; nCr = n!/(r!(n − r)!) unordered ones. With repetition the ordered count is nʳ and the unordered count is C(n + r − 1, r) (stars and bars). Circular arrangements of n distinct items number (n − 1)!, because rotations coincide.
Exact arithmetic
Factorials grow past double precision at 171!; the page computes with arbitrary-precision integers, so 52! is written out in full (68 digits) and nCr is computed as a product of r terms divided as it goes, never overflowing. Scientific notation is only a rounded view.
Reading the numbers
The probability of one particular unordered selection is 1 ÷ nCr; Pascal's row lists nC0 … nCn (up to n = 40). r larger than n gives 0 without repetition and the with-repetition counts still apply. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- nPr = n!/(n − r)!, nCr = n!/(r!(n − r)!), nʳ, C(n + r − 1, r), (n − 1)! for circular arrangements — computed exactly as big integers
Last reviewed 21 September 2026. How results are checked: How we verify.