Permutations vs. Combinations: How to Tell Which One You Need
September 18, 2026
Ask one question: if you swapped the order of two selected items, would the outcome actually be different? If yes, you need a permutation. If no, you need a combination.
Skip the math and get your answer instantly:
Open the Permutation & Combination Calculator →Real-World Examples
| Scenario | Order matters? | Type |
|---|---|---|
| Assigning 1st/2nd/3rd place in a race | Yes | Permutation |
| Choosing 3 books to bring on a trip | No | Combination |
| Setting a 4-digit PIN code | Yes | Permutation |
| Picking 6 lottery numbers | No | Combination |
| Assigning specific roles to team members | Yes | Permutation |
| Selecting a committee of 5 from 20 people | No | Combination |
Worked Example of Each
Permutation: ranking the top 3 finishers out of 10 racers. 10P3 = 10! / (10−3)! = 10 × 9 × 8 = 720 possible ordered outcomes.
Combination: choosing a 3-person committee from 10 people. 10C3 = 10! / (3! × 7!) = 120 possible unordered groups — far fewer, since every group of 3 that could be ranked 720 different ways collapses into a single combination.
Frequently Asked Questions
How do I know if I need a permutation or combination?
Ask whether the order or arrangement of the selected items matters to the result. If yes (ranking, sequencing, assigning distinct roles), use a permutation; if the selection is just a group with no ranking, use a combination.
Why is 10P3 bigger than 10C3?
Because permutations count every possible ordering of the same selected group separately, while combinations count each group only once regardless of order — so nPr is always greater than or equal to nCr for the same n and r.
What's the formula relationship between nPr and nCr?
nPr = nCr × r! — permutations equal combinations multiplied by the number of ways to arrange (order) each selected group.
Are lottery numbers permutations or combinations?
Combinations — the order the winning balls are drawn in doesn't matter, only which specific numbers end up selected.
How many ways can 4 people be seated in a row?
4! = 24 — a permutation, since which seat each person sits in changes the outcome.