Permutations vs. Combinations (nPr vs nCr)
The difference between permutations and combinations, the nPr and nCr formulas, when to use each, and worked examples.
5 min read · Reviewed August 2026
The key difference: does order matter?
Permutations and combinations both count the ways to choose r items from n, but they answer different questions. The single rule that separates them is whether order matters.
- Permutation (nPr) — order matters. “ABC” and “CBA” are different. Think arrangements, rankings, passwords, podium finishes.
- Combination (nCr) — order does not matter. “ABC” and “CBA” are the same. Think groups, committees, lottery numbers, pizza toppings.
The formulas
Permutations divide out only the items you did not choose; combinations also divide out the orderings of the ones you did:
Because combinations ignore order, nCr is always smaller than or equal to nPr — in fact nCr = nPr ÷ r!.
Worked examples
From 5 people, how many ways can you choose 2?
- As a ranked pair (1st and 2nd place) — order matters → 5P2 = 5! / 3! = 20.
- As an unordered team of two — order does not matter → 5C2 = 5! / (2! · 3!) = 10.
There are exactly twice as many permutations as combinations here, because each unordered pair can be arranged in 2! = 2 ways.
A quick decision rule
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters; a combination counts selections where order does not. For the same n and r, there are always at least as many permutations as combinations.
When do I use nPr instead of nCr?
Use nPr when the order of the chosen items matters — arrangements, rankings, sequences, passwords. Use nCr when only the group matters — teams, committees, lottery draws.
Why is nCr smaller than nPr?
Because combinations treat every ordering of the same items as one outcome. Dividing the permutation count by r! (the number of ways to order r items) gives the combination count.