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Permutations (nPr)

Also known as nPr · ordered arrangements

nPr=n!(n−r)!{}_{n}P_{r} = \frac{n!}{(n-r)!}

Units aren’t used in this calculation — every value is a plain number.

Worked example: 10P3 = 720 podiums — press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

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Permutations (nPr) explained

nrnPr

A permutation is a selection where order matters — gold, silver and bronze are three different results, not one. With ten sprinters in a final there are 10P3 = 10!/7! = 10 × 9 × 8 = 720 possible podiums. The formula is just the falling product n(n−1)…(n−r+1) written as a ratio of factorials, since dividing by (n − r)! cancels the tail you never reach.

Note that there is no closed-form way to run this backwards, so this calculator solves only for the count. The classic trap is choosing permutations where combinations belong: a five-card poker hand is not 52P5 = 311,875,200, because the same five cards dealt in a different order are the same hand — divide by 5! to get 2,598,960. Ask yourself whether rearranging the chosen items produces a genuinely different outcome. If yes, permutations; if no, combinations. A four-letter code from 26 distinct letters with no repeats allows 26P4 = 26 × 25 × 24 × 23 = 358,800 codes.

Permutations (nPr) formula

nPr=n!(n−r)!{}_{n}P_{r} = \frac{n!}{(n-r)!}
Where
  • nPr{}_{n}P_{r}= Number of ordered selections
  • nn= Items available
  • rr= Items chosen

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