Permutations and Combinations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Order matters, or it does not

Mathematics · Discrete Mathematics · ages 18-19
Name ______________________   Date ____________

  1. Five runners race for gold and silver medals. How many outcomes are possible?

    Answer: ______________

  2. A pizza shop offers 5 toppings. How many ways can you choose 2 toppings?

    Answer: ______________

  3. A club selects 3 members with equal duties to form a committee. Is this a permutation or a combination?

    • A combination, since order does not matter
    • A permutation, since order matters
    • Neither, since the sum rule applies

  4. Swapping two committee members with equal duties gives a different committee.

    Circle one:   True   False

  5. How many distinct arrangements are there of the letters in LEVEL?

    • 30
    • 60
    • 120
    • 15

  6. How many distinct arrangements do the letters of LEVEL have?

    • 30
    • 120
    • 60

  7. How many 2-person teams can form from 7 players? Type the number.

    Answer: ______________

  8. P(7,2) = 42 counts ordered pairs from 7 items. Divide out the repeats to get C(7,2). What is it?

    Answer: ______________

  9. Ben counts 3-person committees from 10 students as 10 times 9 times 8, or 720. What is wrong?

    • He divided by 6 for no reason; 720 is right
    • He counted ordered selections but committees ignore order; divide by 6 to get 120
    • He should have added the counts; the total is 27

  10. How many distinct arrangements are there of the letters B, A, N, A, N, A?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Order matters, or it does not W1-mt_1L58kuu-w3-s1

  1. 20 · Medal order matters, so P(5,2) = 5 by 4 = 20.
  2. 10 · Order does not matter, so C(5,2) = 10.
  3. A combination, since order does not matter · Equal duties mean reordering changes nothing, so this is a combination.
  4. False · Equal members stay the same committee however you list them.
  5. 30 · 5! over (2! by 2!) = 120 over 4 = 30.
  6. 30 · LEVEL repeats E and L, so divide 120 by 2 factorial twice: 30.
  7. 21 · Choosing 2 from 7 is 7 times 6 divided by 2, which is 21.
  8. 21 · Each pair was counted twice, so 42 over 2 = 21.
  9. He counted ordered selections but committees ignore order; divide by 6 to get 120 · Ben ran a permutation count on a combination problem, so he must divide out the 6 orders.
  10. 60 · 6! over (3! by 2!) = 720 over 12 = 60.
Worksheet · LightMySky