Combinations Calculator (nCr)
Find how many ways you can choose r items from a set of n when the order of selection does not matter.
Ways to choose r items when order does not matter
120
of 2^n subsetsSelected combinations
11.7%
Other subsets
88.3%
- 1
Smaller half k = min(r, n−r)
min(3, 7) = 3 - 2
Falling factorial (numerator)
10 × 9 × 8 = 720 - 3
k! (denominator)
1 × 2 × 3 = 6 - 4
C(n, r) = numerator ÷ k!
720 ÷ 6 = 120
How does this calculator work?
A combination C(n, r) = n!/(r!·(n−r)!) counts how many ways you can choose r items from n when order does not matter. Enter n and r (whole numbers, 0 ≤ r ≤ n) to get the exact count, plus its share of all 2^n possible subsets and the related permutation total.
Formula
How this is calculated
Enter the total number of items n and how many you want to choose r. Both must be whole numbers with 0 ≤ r ≤ n. A combination counts each unordered selection once, so picking {A, B} is the same as {B, A} — this is what distinguishes combinations from permutations.
The value is computed with the multiplicative formula C(n, r) = ∏(n − k + i)/i for i = 1…k, using k = min(r, n − r). This avoids computing huge factorials directly and keeps intermediate values small, so it stays accurate for moderately large n. The donut compares C(n, r) against the total number of subsets of an n-element set, which is 2^n, showing what fraction of all possible subsets have exactly r elements.
Edge cases follow convention: C(n, 0) = C(n, n) = 1, and C(n, r) = 0 conceptually when r > n (the calculator asks for valid input instead). Results are exact integers for small inputs; for very large n the 2^n total and permutation count may exceed safe precision and are flagged accordingly.
Frequently asked questions
Combinations (nCr) count selections where order does not matter, while permutations (nPr) count ordered arrangements. nPr = nCr × r!, so there are always at least as many permutations as combinations.
There is exactly one way to choose nothing from a set — the empty selection. By the same logic C(n, n) = 1, since there is one way to choose all items.
It is the binomial coefficient: the coefficient of the x^r term when expanding (1 + x)^n, also written as "n choose r".
Also known as
TG we-Calculate Editorial Team. (2026). Combinations Calculator (nCr) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/combinations-calculator
TG we-Calculate Editorial Team. "Combinations Calculator (nCr)." TG we-Calculate. 2026. https://we-calculate.com/calculator/combinations-calculator.
TG we-Calculate Editorial Team, "Combinations Calculator (nCr)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/combinations-calculator
@misc{wecalculate_combinations_calculator, title = {Combinations Calculator (nCr)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/combinations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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