THE NUMORIX GUIDE
How to use the Combination Calculator
Last reviewed September 14, 2026
What this calculator does
For selecting r items from n without regard to order, use C(n,r) = n! / (r!(n-r)!).
Formula and method
For selecting r items from n without regard to order, use C(n,r) = n! / (r!(n-r)!).
Variables and inputs
n is total items and r is selected items. The UI defaults to n=10 and r=3 and requires nonnegative integers with r <= n.
Worked example
For n=10, r=3: C(10,3) = 10!/(3!*7!) = (10*9*8)/(3*2*1) = 120.
How to interpret the result
The result counts distinct groups, so selecting A then B is the same group as selecting B then A.
Common mistakes to avoid
Do not use n^r for combinations, and do not divide by r! unless you started with an ordered count.
Assumptions and limitations
The engine uses Number factorials, so sufficiently large n can overflow or lose precision. It does not model repeated or indistinguishable items.
Practical use and checks
Combinations count groups when order does not matter. Use the calculator for a committee, a set of lottery numbers, or a sample selected without positions. Enter n = 10 and r = 3 as a check; the result should be 120 from 10! divided by 3! times 7!. Selecting Alice, Ben, and Chen is the same group as selecting Chen, Alice, and Ben, which is why the answer is smaller than the permutation result of 720. Before entering values, decide whether items are distinct and whether selection is with or without replacement. Repeated product types, repeated lottery numbers, or a committee with role assignments can require a different model. The identity C(n,r) = C(n,n-r) offers a useful check: choosing 3 of 10 should equal choosing the 7 left out. The calculator expects nonnegative integers with r no greater than n. Very large factorials can exceed JavaScript Number precision in implementations that format through ordinary numbers, so an apparently neat answer may not be exact at high n. C(n,0) and C(n,n) are both 1, while invalid denominator cases should not be treated as meaningful counts.