Permutation Calculator
Compute the number of ordered arrangements of r items chosen from n distinct
items — written nPr. Perfect for questions about rankings,
finishing orders, prize distributions and password possibilities where the
order of selection matters.
Enter n and r
Enter n and r, then press Calculate nPr.
The permutation formula
nPr = n! / (n − r)!
Equivalently, nPr = n × (n − 1) × (n − 2) × … × (n − r + 1) —
the product of the first r integers counting down from n.
Why? Choosing an ordered arrangement of r items from n means: pick any of the n items for the first position, any of the n − 1 remaining for the second, and so on until r positions are filled. Multiplying those choices gives the formula above.
Permutation vs combination
Permutations count order-sensitive selections.
Combinations (nCr) count order-insensitive selections — for
instance, which three people are on a committee, not who is first, second
and third.
nPr = nCr × r!
Worked examples
Example 1 — Podium finishes
10 runners compete. How many ways can gold, silver and bronze be awarded?
10P3 = 10 × 9 × 8 = 720.
720 arrangements
Example 2 — Password possibilities
How many 4-digit codes can be made from digits 0–9 if repetition is not allowed?
10P4 = 10 × 9 × 8 × 7 = 5040.
5040 codes
Example 3 — Permuting all items
How many ways can 5 books be arranged on a shelf?
5P5 = 5! = 120.
120 arrangements
Related calculators
Frequently asked questions
nPr counts arrangements where order matters. nCr counts selections where order does not matter. For the same n and r, nPr = nCr × r! — because every set of r items can be arranged in r! different orders.
0! = 1 by convention — there is exactly one way to arrange nothing. This definition keeps combinatorial formulas like nPr = n! / (n − r)! valid even when r = n.
If r = 0, there is 1 way to choose nothing: nP0 = 1. If r = n, all n items are arranged, so nPn = n!.