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A permutation is an ordered arrangement of items. When the order matters — such as ranking 3 winners from 10 contestants — you use the permutation formula nPr (also written P(n,r)).
Formula: nPr = n! / (n − r)! where n is the total number of items and r is how many you are choosing. For example, P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720 ordered arrangements.
Permutations differ from combinations in one key way: order matters. The arrangement (A, B, C) counts separately from (B, A, C) in permutations, but they count as the same selection in combinations. Use permutations for ranking problems, passwords, race finishing positions, and any scenario where sequence is significant.
Worked example — how many ways can 4 books be arranged on a shelf from a collection of 9? P(9, 4) = 9! / (9−4)! = 9! / 5! = 9 × 8 × 7 × 6 = 3,024 arrangements.
nPr = n! / (n − r)!
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.