Enter n (total objects) and r (sample size) to calculate the number of possible combinations (nCr) — selections where order doesn't matter. Also get the permutations (nPr) count and factorial values side by side, so you can compare both results at once.
Combinations C(n,r)
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Permutations P(n,r)
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n! (n Factorial)
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r! (r Factorial)
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(n−r)! Factorial
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A combination is a selection of r items chosen from a set of n items where the order of selection does not matter. For example, choosing 2 fruits from {apple, banana, cherry} gives the same combination whether you pick apple then banana, or banana then apple.
The formula is C(n, r) = n! / (r! × (n − r)!), where n is the total number of objects and r is the sample size. The exclamation mark denotes factorial — the product of all positive integers up to that number.
Combinations count selections where order does NOT matter, while permutations count arrangements where order DOES matter. Permutations (nPr) = n! / (n − r)!, which is always greater than or equal to the number of combinations for the same n and r.
No — you cannot choose more items than exist in the set. If r > n, the combination is undefined (or mathematically zero) because you cannot form valid subsets. This calculator will flag that as an invalid input.
Without replacement means each item can only be selected once. Once an item is chosen, it is removed from the pool. This is the standard assumption for C(n, r) and is what this calculator computes.
If n people all shake hands with each other exactly once, the total number of handshakes equals C(n, 2) = n(n−1)/2. For example, 6 people produce C(6, 2) = 15 unique handshakes, because each pair is counted only once.
A factorial (n!) is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials form the building blocks of the combinations formula — specifically n!, r!, and (n−r)!.
Simply enter the total number of objects (n) and the sample size (r), then the calculator instantly shows the number of combinations C(n,r) and permutations P(n,r) along with the relevant factorials. Results update automatically as you change the inputs.