Calculate combinations nCr and permutations nPr with or without repetition, factorial expansions, and subset ordering.
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A permutations and combinations calculator determines the number of distinct ways to choose and arrange r items from a collection of n total items. Combinations apply when selection order does not matter (e.g. lottery balls, poker hands), while permutations apply when order is critical (e.g. race rankings, security pin codes).
nPr = \frac{n!}{(n-r)!}, \quad nCr = \binom{n}{r} = \frac{n!}{r!(n-r)!}, \quad \text{Repetition: } n^r \text{ and } \binom{n+r-1}{r}Where n is total items, r is items chosen, and n! denotes the factorial product of all positive integers from 1 up to n.
Scenario: Find how many unique committees of 3 people can be selected from an 8-person team.
Set n = 8 (total candidates) and r = 3 (seats on committee).
Because committee roles are equal, order does not matter â use combinations nCr.
nCr = 8! / (3! Ă 5!) = (8 Ă 7 Ă 6) / (3 Ă 2 Ă 1) = 336 / 6 = 56.
If assigning distinct roles (President, VP, Treasurer), use permutations: nPr = 8! / 5! = 336.
Use combinations when rearranging the chosen items produces the exact same group (e.g., a hand of playing cards or a team). Use permutations when order creates a new outcome (e.g., a locker combination lock or running race results).
Because entering the numbers 12-34-56 in the order 56-12-34 will not open the lock! The order of entry is essential, making it mathematically a permutation.
Repetition means an item can be selected more than once (e.g. creating a 4-digit PIN where digits 0000 or 1212 are allowed). Without repetition, each chosen item is removed from the remaining pool.
0! = 1 by mathematical convention and algebraic necessity. There is exactly one way to arrange zero objects: the empty set.
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