Combination Calculator
Count selections of distinct items when the order of the selected items does not matter.
Combinations (order does not matter)
Distinct items, no repetition, and 0 ≤ r ≤ n ≤ 10,000. Choosing no items has one outcome, the empty selection. Exact output is limited to 2,000 digits.
Formula checked · NumberSwift · Review process
How this calculator works
A combination treats the same chosen set as one selection regardless of its arrangement. Choosing three people for a committee therefore differs from assigning three different positions. The implementation multiplies and divides exact integers progressively rather than forming floating-point factorials. The identity nCr = nC(n−r) shortens the calculation. Very large results remain exact up to the stated output limit.
Formula
nCr = n! / (r!(n−r)!). Equivalently, multiply (n−i+1)/i for i = 1…min(r,n−r), keeping integer intermediates.
Assumptions & limitations
Distinct items, no repetition, and 0 ≤ r ≤ n ≤ 10,000. Choosing no items has one outcome, the empty selection. Exact output is limited to 2,000 digits.
Worked example
Choosing 3 items from 10 gives 10!/(3!7!) = (10 × 9 × 8)/(3 × 2 × 1) = 120 combinations.
Our approach to calculations ↗Common questions
When should I use permutations instead?
Use Permutation when order or assigned position matters. A committee is a set; first, second and third place are ordered positions.
Why does choosing zero items return one?
There is exactly one empty selection. This is the standard combinatorial convention.
Will a large result be rounded?
The integer count is exact within the 2,000-digit output limit. Unsupported inputs show a clear limit instead of Infinity.