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Problems involving both permutations and combinations

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Permutation or Combination?

Deciding between a permutation and a combination comes down to one question: does order matter? Learn the rule, see it applied to permutations (nPr) and combinations (nCr), and work through mixed problems that use both in stages.

The one question that decides it

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.

Choosing between a permutation and a combination Ask: does order matter? If yes, use a permutation (nPr) -- for example choosing a president and vice-president from 6 people, where 6P2 = 30. If no, use a combination (nCr) -- for example choosing a 2-person committee from the same 6 people, where 6C2 = 15. Does ORDER matter? (president vs. VP, or just a group?) YES NO Permutation: nPr Choose president & VP from 6 people: 6P2 = 6·5 = 30 Combination: nCr Choose a 2-person committee from 6: 6C2 = 30/2! = 15
Ask whether order matters, then pick a permutation or 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).

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