CalcLibrary

Permutations & Combinations Calculator

Choose whether order matters and whether repetition is allowed, then get the exact permutations or combinations count with the matching formula.

Examples

Order changes the count because gold, silver, and bronze are different positions.

Total outcomes
60
Selected counting mode
Ordered arrangement without repetition
Formula in symbols
P(n,r) = \frac{n!}{(n-r)!}
With your values
\frac{5!}{(5-3)!} = \frac{5!}{2!} = 60

Order matters here: AB and BA count as different outcomes, and each item can appear at most once.

Was this useful?

Examples

How It Works

Formula

P(n,r)=n!(n−r)!P(n,r) = \frac{n!}{(n-r)!}

Prep(n,r)=nrP_{\text{rep}}(n,r) = n^r

C(n,r)=n!r!(n−r)!C(n,r) = \frac{n!}{r!(n-r)!}

Crep(n,r)=(n+r−1)!r!(n−1)!C_{\text{rep}}(n,r) = \frac{(n+r-1)!}{r!(n-1)!}

Variables

nn

Number of distinct available items

rr

Number of positions or picks being filled

Start with the two yes/no decisions. If order matters, use a permutation formula. If order does not matter, use a combination formula. Then decide whether each item can be used once or reused, and the calculator switches to the matching one of the four standard counting cases.

Use the toggle pair as a 2-by-2 decision table:

| Order matters? | Repetition allowed? | Use | | --- | --- | --- | | Yes | No | P(n,r)=n!(n−r)!P(n,r)=\frac{n!}{(n-r)!} | | Yes | Yes | Prep(n,r)=nrP_{\text{rep}}(n,r)=n^r | | No | No | C(n,r)=n!r!(n−r)!C(n,r)=\frac{n!}{r!(n-r)!} | | No | Yes | Crep(n,r)=(n+r−1)!r!(n−1)!C_{\text{rep}}(n,r)=\frac{(n+r-1)!}{r!(n-1)!} |

The calculator keeps the arithmetic exact with integer counting logic, then shows the symbolic formula, your substituted values, and the final count.

Frequently Asked Questions

01When should I use permutations instead of combinations?
Use permutations when position changes the outcome. If ABC and BAC should count separately, order matters and you need a permutation formula.
02What does repetition mean here?
Repetition means the same item can be chosen again. A PIN code allows repetition; a podium of distinct winners does not.
03Why is r > n blocked in some modes?
Without repetition, you cannot choose more distinct items than exist. If you need more picks than the pool contains, repetition must be allowed.
04What happens when r = 0?
There is exactly one way to choose nothing: the empty arrangement or empty selection. That is why the result is 1 when r = 0.
05Why do the answers grow so quickly?
Counting problems multiply choices across positions. Even modest values of n and r can produce very large exact integers.

All calculators