Permutations & Combinations

Permutations, combinations, sampling with repetition, factorials, derangements and Catalan numbers, exact to the last digit

n · total items

k · items picked

Results

PermutationsP(10, 3)

n! / (n − k)!

720

CombinationsC(10, 3)

n! / (k! × (n − k)!)

120

Permutations with repetition10^3

n^k

1,000

Combinations with repetitionC(10 + 3 - 1, 3)

C(n + k − 1, k)

220

Factorial10!

3,628,800

Derangements!10

!n = (n − 1)(!(n − 1) + !(n − 2))

1,334,961

Catalan numberC(20, 10) / 11

C(2n, n) / (n + 1)

16,796

Stirling number (2nd kind)S(10, 3)

S(n, k) = k × S(n − 1, k) + S(n − 1, k − 1)

9,330

Bell numberB(10)

B(n) = S(n, 0) + S(n, 1) + … + S(n, n)

115,975

Both n and k are non-negative integers. When k is larger than n only the ordered and unordered selection rows have no answer and show —; factorials, derangements and the rest still compute. n is capped at 2000, Stirling and Bell numbers at 200.

Everything is exact BigInt arithmetic, no floating point. Values longer than 24 digits switch to scientific notation in the table, while the copy button always yields the full number. The math never leaves the browser.