Permutation or Combination?

Many problems mix permutations and combinations in the same setup, and the fastest way through is a single question: does the order of selection matter? If rearranging the same people or items creates a different outcome, you need a permutation. If the outcome only depends on who or what is selected, you need a combination.

Assigning distinct roles — a president and a vice-president, first place and second place — means swapping two people changes the result. That is a job for permutations: choosing a president and vice-president from 6 people gives 6P2 = 6 × 5 = 30 ordered outcomes.

Choosing a group with no distinct roles — a 2-person committee, a hand of cards — means the same people in a different order are the same outcome. That is a job for combinations: choosing a 2-person committee from the same 6 people gives 6C2 = 30 ÷ 2! = 15 outcomes, exactly half the permutation count, because each pair of people can be ordered 2 ways.

Many real problems combine both. For example, choosing a 3-person committee from 8 people (combination, since committee membership has no order), then selecting one of those 3 as chair (a further ordered choice) requires applying the rule twice: 8C3 for the committee, then 3 for the chair choice. Break a mixed problem into stages and ask the order question at each stage.

• Permutation: arranging 4 books in a specific order on a shelf.
• Combination: choosing 4 books to bring on a trip (no order).
• Mixed: selecting 3 finalists from 10 (combination), then ranking them gold/silver/bronze (permutation).

Related lessons

• Statistics