Possible Combinations Calculator — nCr With & Without Replacement
Find how many distinct ways you can choose r items from a group of n, with or without repetition. Enter n, r and the replacement rule — the calculator applies the correct formula and shows the result instantly.
Replacement rule
Unordered selections of r from n, each item used at most once
- 1
Items not chosen (n − r)
10 − 3 = 7 - 2
C(n, r) = n! ÷ (r! × (n−r)!)
10! ÷ (3! × 7!) = 120
How does this calculator work?
C(n, r) = n! / (r! × (n−r)!) counts unordered selections without repetition; C(n+r−1, r) counts them with repetition. Enter pool size n and selection size r — the calculator handles both modes, shows the formula applied, and uses an iterative algorithm to handle large numbers.
Formula
How this is calculated
A combination counts unordered selections — choosing a committee of 3 from 10 people, for example, where the order does not matter. The standard formula C(n, r) = n! / (r! × (n−r)!) counts every distinct subset of size r that can be drawn from a pool of n without reusing any item. The denominator r! cancels out all orderings of the chosen items (since selecting Alice, Bob, Carol is the same committee as Carol, Alice, Bob), and (n−r)! cancels the items never chosen.
With replacement (repetition allowed) each item can appear more than once in a selection — like choosing ice-cream scoops when you can repeat a flavour. The formula becomes C(n+r−1, r), known as the "stars and bars" result. This counts the number of ways to distribute r indistinguishable choices among n distinct types, and gives a larger result than the without-replacement case for the same n and r.
Both formulas assume order is irrelevant — choosing ABC is the same as BAC. If order matters, you want permutations instead: nPr = nCr × r!. The calculator uses an iterative algorithm to avoid floating-point overflow for moderate inputs, but for very large n the results may become approximate.
Frequently asked questions
Permutations (nPr) count ordered selections — every different ordering of the same items is counted separately. Combinations (nCr) treat different orderings as identical. nPr = nCr × r!, so nCr is always ≤ nPr.
Use the with-replacement formula when the same item can be chosen more than once, such as choosing 3 flavours of ice cream from 5 options while being allowed to pick the same flavour twice. The formula C(n+r−1, r) counts these multiset selections.
You cannot choose more distinct items than exist in the pool. If r > n the without-replacement formula is undefined (the factorial (n−r)! of a negative number is undefined). The with-replacement formula has no such restriction since items can repeat.
Also known as
TG we-Calculate Editorial Team. (2026). Possible Combinations Calculator — nCr With & Without Replacement [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/possible-combinations-calculator
TG we-Calculate Editorial Team. "Possible Combinations Calculator — nCr With & Without Replacement." TG we-Calculate. 2026. https://we-calculate.com/calculator/possible-combinations-calculator.
TG we-Calculate Editorial Team, "Possible Combinations Calculator — nCr With & Without Replacement," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/possible-combinations-calculator
@misc{wecalculate_possible_combinations_calculator, title = {Possible Combinations Calculator — nCr With & Without Replacement}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/possible-combinations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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