Permutation vs Combination: Key Differences Explained
Compare permutations and combinations to know when order matters in counting problems and probability calculations.
Quick Answer
Use permutations when order matters; use combinations when it does not.
| Feature | Permutation | Combination |
|---|---|---|
| Order matters (AB is different from BA) | Order does not matter (AB = BA) | |
| More arrangements possible | Fewer groupings possible | |
| Formula: n! / (n-r)! | Formula: n! / [r!(n-r)!] | |
| Example: ranking, passwords, race placements | Example: lottery, teams, committees |
Permutations count the number of ways to arrange items where order matters. Choosing a president, vice president, and secretary from 10 people is a permutation problem because the roles are distinct.
Combinations count the number of ways to choose items where order does not matter. Selecting 3 people for a committee from 10 is a combination problem because the group is the same regardless of selection order.
When to Use Permutation
- Arranging items in a specific sequence
- Calculating password or PIN possibilities
- Determining race finish orders
When to Use Combination
- Selecting a group or committee
- Calculating lottery odds
- Choosing items from a set without regard to order
Worked Example
Choose 3 people from a group of 10.
Permutation
Permutations (order matters): 10 x 9 x 8 = 720 arrangements.
Combination
Combinations (order doesn't matter): 720 / 3! = 120 groups.
There are 6x more permutations because each group of 3 can be arranged in 3! = 6 ways.
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Frequently Asked Questions
How do I know if order matters?
Ask: would rearranging the selected items create a different outcome? If yes, use permutations.
Is a lottery a permutation or combination?
Most lotteries are combinations — the order you pick numbers does not matter.
Why are there always more permutations?
Because permutations count every arrangement of the same group as a different outcome.