Combinations
Choosing a group when the order does not matter. How C(n, r) = n! / (r!(n-r)!) works, why dividing by r! cancels the repeated orderings, and how combinations differ from permutations.
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Choosing a group when the order does not matter. How C(n, r) = n! / (r!(n-r)!) works, why dividing by r! cancels the repeated orderings, and how combinations differ from permutations.
A combination is a way of choosing a group from a larger set when the order does not matter. Picking Ahmed, Sara, and Mohammed for a team is the same choice however you list them.
A combination chooses elements from with no regard to order, so and are the same combination — counted once.
Use combinations whenever you pick a group — a team, a committee, a hand of cards — where it does not matter who was chosen first.
Five teams, each pair playing once. Order does not matter, so count pairs:
A committee of 4 from a class of 12
Choosing 3 salads from 8 kinds
A lineup of 5 players from 15
| Property | Combinations C(n, r) | Permutations P(n, r) |
|---|---|---|
| Order | does not matter | matters |
| Formula | n! ÷ (r!(n − r)!) | n! ÷ (n − r)! |
| Result | smaller (or equal) | larger (or equal) |
| Example | choosing a team of 3 | ranking 3 in positions |
| C(5, 2) vs P(5, 2) | 10 | 20 |