If the order of the chosen items changes the outcome, use a permutation (nPr). If it does not, use a combination (nCr). The swap test below makes this decision before any numbers are written.
This lesson opens the permutations and combinations chapter. Later lessons use it in restricted arrangements and selections.
What is the swap test?
Imagine one outcome and swap two of the chosen items. Ask whether you now have a different outcome or the same one. A president and a secretary swapped make a different outcome, so order matters. Two committee members swapped are still the same committee, so order does not matter.
Worked example: the same eight students
Eight students are available.
Case A. Choose a president, a secretary and a treasurer. Swapping the secretary and treasurer gives a different result, so order matters:
8P3 = 8 × 7 × 6 = 336
Case B. Choose a committee of 3 with no roles. Swapping two members gives the same committee, so order does not matter:
8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 56
The link: 336 = 56 × 6. Each committee of 3 can be given roles in 3! = 6 ways, so 56 committees produce 336 role assignments.
A second pair: codes and teams
A four-digit code uses digits from 1 to 9 with no repeats. The code 1234 is different from 4321, so this is 9P4 = 9 × 8 × 7 × 6 = 3 024.
A team of 4 chosen from 9 players has no order, so this is 9C4 = 3 024 ÷ 24 = 126.
The mistake that costs marks
A common error is using nCr for a question that assigns roles, or nPr for a question that only chooses a group. The numbers look reasonable either way, so the error is hard to see.
A sense check helps. If your answer to a roles question is smaller than the matching selection count, something is wrong, since nPr is at least as large as nCr.
Check yourself
Ten runners are in a race. (a) In how many ways can gold, silver and bronze go to three different runners? (b) In how many ways can 3 of the 10 runners be chosen for a workshop?
Answer
(a) The medals differ, so order matters: 10P3 = 10 × 9 × 8 = 720.
(b) A workshop group has no order: 10C3 = 720 ÷ 6 = 120.
Check the link: 120 × 3! = 720, so the two answers agree.
What to study next
Continue with counting arrangements with restrictions, where some items must sit together or in fixed places. The word-problem structure worksheet helps you identify the roles in a long question.
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