Permutations and combinations count how many ways you can pick r things from n. The difference is order. A permutation treats different orders as different outcomes: nPr = n!/(n−r)!. A combination ignores order, counting only which items are chosen: nCr = n!/(r!(n−r)!). Since each chosen group can be arranged in r! ways, nPr is always r! times nCr. The single question to ask is simply: does rearranging the selection make it a new outcome?
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the standard nPr and nCr definitions.
The counting formulas
A factorial counts the arrangements of n distinct items. The permutation formula cancels the (n−r)! arrangements of the items you didn't pick, leaving the ordered selections of the r you did. The combination formula then divides by r! to remove the orderings within the chosen group, since for a combination those count as one. That r! factor is exactly why nPr ≥ nCr, with equality only when r ≤ 1.
Worked example — choose 2 from 5
Scenario: n = 5 items, choose r = 2.
There are 20 ordered ways and 10 unordered ways to pick 2 of 5 — and indeed 20 = 2! × 10, the r! relationship. The pattern scales: choosing 3 of 10 gives 720 permutations and 120 combinations; choosing all 6 of 6 gives 6! = 720 permutations but only 1 combination (there's just one way to take everything). And a 5-card hand from a 52-card deck is nCr(52,5) = 2,598,960 — the foundation of poker odds.
Frequently Asked Questions
Permutations count order, combinations don't. 5 choose 2 → 20 vs 10.
n!/(n−r)!. 5,2 → 120/6 = 20. Or multiply the top r terms: 5×4 = 20.
n!/(r!(n−r)!) = nPr/r!. 5,2 → 120/12 = 10. nCr(52,5) = 2,598,960.
n! = n×(n−1)×…×1. 5! = 120. 0! = 1. Grows very fast.
Order matters → nPr (codes, race order). Only the group → nCr (lottery, hands).