Combination vs. Permutation
A combination counts selections where order does not matter — like picking 3 people for a committee. A permutation counts arrangements where order does matter — like assigning 1st, 2nd, and 3rd place. For the same and , there are always more permutations than combinations because each combination can be rearranged in ways.
Combination vs. Permutation
| Combination | Permutation | |
|---|---|---|
| Definition | A selection of items where order does NOT matter | An arrangement of items where order DOES matter |
| Formula | ||
| Order matters? | No | Yes |
| Example | Choosing 3 students from 10 for a committee | Choosing president, VP, and secretary from 10 candidates |
| C(10,3) vs P(10,3) | 120 | 720 |
| Relationship | ||
| Also called | n choose r, binomial coefficient | n permute r, arrangement |
When to Use Each
Use Combinations when...
- Choosing a group, team, or committee
- Selecting items from a menu (order doesn't matter)
- Lottery numbers (any order wins)
- Poker hands (suit combinations, not sequence)
Use Permutations when...
- Arranging items in a specific order
- Assigning ranks or positions (1st, 2nd, 3rd)
- Creating passwords or codes (character order matters)
- Scheduling events in a sequence
Examples
Combination example
How many ways can you choose 2 fruits from {apple, banana, cherry}? The answer is : {apple, banana}, {apple, cherry}, {banana, cherry}. Notice {banana, apple} is the same as {apple, banana}.Permutation example
How many ways can you arrange 2 of those fruits in a row? The answer is : (apple, banana), (banana, apple), (apple, cherry), (cherry, apple), (banana, cherry), (cherry, banana). Now order matters, so we get twice as many.Common Confusion Points
The most common mistake is using a permutation formula when order doesn't matter (or vice versa). Ask yourself: would rearranging the selected items create a different outcome? If yes → permutation. If no → combination.
Another common error: confusing with . These are actually equal — choosing which items to include is the same as choosing which to leave out.
Frequently Asked Questions
Is a combination always smaller than a permutation?
Yes, for the same n and r (with r > 1), C(n,r) < P(n,r). This is because each combination corresponds to r! permutations — the number of ways to rearrange the selected items.
When do combinations and permutations give the same answer?
When r = 0 or r = 1. C(n,0) = P(n,0) = 1, and C(n,1) = P(n,1) = n. With 0 or 1 item, there's nothing to rearrange.
What does 'n choose r' mean?
'n choose r' is another way to say C(n,r) — the number of combinations of r items chosen from n. It is written as the binomial coefficient (n above r in parentheses) and appears throughout probability, algebra, and the Binomial Theorem.