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Permutation and Combination Calculator
Determine the number of permutations and combinations for a given set size (n) and subset size (r), with and without repetition.
Choose n and r
Results
| Permutations (Order Matters, No Repetition) — P(n, r) | — |
|---|---|
| Combinations (Order Doesn't Matter, No Repetition) — C(n, r) | — |
| Permutations with Repetition — n^r | — |
| Combinations with Repetition — C(n+r-1, r) | — |
Permutations vs. Combinations
Permutations and combinations calculate the number of unique groupings that can be made from a set of objects. The main distinction between them lies in whether order matters.
Permutations (Order Matters)
Use permutations when the arrangement order of the chosen items is important. For example, arranging books on a shelf, selecting a President and a Vice President, or setting a passcode.
**Formula without repetition:** $$P(n, r) = \frac{n!}{(n-r)!}$$ **Formula with repetition:** $$P_R(n, r) = n^r$$
Combinations (Order Does Not Matter)
Use combinations when the arrangement order does not matter; only the grouping is important. For example, picking lottery numbers, choosing toppings for a pizza, or selecting a committee of members with equal roles.
**Formula without repetition:** $$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$ **Formula with repetition:** $$C_R(n, r) = \binom{n+r-1}{r} = \frac{(n+r-1)!}{r!(n-1)!}$$