Permutations vs Combinations: Formulas, Differences, and Examples
Understand when order matters, learn the nPr and nCr formulas, and compare permutations and combinations with clear examples.
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Does order matter?
A permutation counts arrangements where order matters. A combination counts selections where order does not matter. Choosing a gold, silver, and bronze medalist is a permutation. Choosing a three-person committee is a combination.
Formulas
Here n is the total number of items and r is how many you choose.
- Factorial: n! = n × (n - 1) × ... × 1, and 0! = 1
- Permutations: nPr = n! / (n - r)!
- Combinations: nCr = n! / (r! × (n - r)!)
Example: choosing 3 from 10 people
For medals, 10P3 = 10 × 9 × 8 = 720 ways. For a committee, 10C3 = 720 / 3! = 720 / 6 = 120 ways. The combination count is smaller because each group of 3 people is counted once, not in every order.
| Situation | Type | Result |
|---|---|---|
| Gold, silver, bronze from 10 runners | Permutation (10P3) | 720 |
| Committee of 3 from 10 people | Combination (10C3) | 120 |
Quick tips
- Ask: would swapping two chosen items create a different outcome? If yes, use permutations.
- nPr = nCr × r!, so permutations are always at least as large as combinations.
- Factorials grow very fast, so use a calculator for larger n.
FAQ
What is the difference between nPr and nCr?
nPr counts ordered arrangements, while nCr counts unordered selections.
Why is 0! equal to 1?
By definition, there is exactly one way to arrange zero items, which keeps the formulas consistent.
Can r be larger than n?
No. You cannot choose more items than you have, so nPr and nCr are not defined when r is larger than n.
