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

Count the union without double counting

Mathematics · Discrete Mathematics · ages 19-20
Name ______________________   Date ____________

  1. 30 pupils join the chess club, 25 join the drama club, and 10 join both. How many distinct pupils join at least one club?

    • 45
    • 55
    • 65
    • 35

  2. 30 pupils join the chess club, 25 join drama, and 10 join both. How many distinct pupils join at least one club?

    • 55
    • 45
    • 35

  3. 20 like tea, 15 like coffee, 12 like juice. 8 like tea and coffee, 6 like tea and juice, 5 like coffee and juice, and 3 like all three. How many like at least one? Give a number.

    Answer: ______________

  4. Sam says that in the two-set formula, shared members are counted twice and then subtracted once. Is Sam right?

    Circle one:   True   False

  5. Three letters are placed into three addressed envelopes at random. In how many placements does no letter reach the right envelope?

    • 2
    • 1
    • 3
    • 6

  6. In a survey, 20 like tea, 15 like coffee, 12 like juice. 8 like tea and coffee, 6 like tea and juice, 5 like coffee and juice, and 3 like all three. How many like at least one? Give a number.

    Answer: ______________

  7. How many derangements of 4 items are there? Give a number.

    Answer: ______________

  8. Three letters are placed into three addressed envelopes at random. In how many placements does no letter reach the right envelope?

    • 2
    • 1
    • 6

  9. You extend inclusion-exclusion to four sets. What comes right after adding back the triple overlaps?

    • Add the triple overlap a second time
    • Subtract the quadruple overlaps
    • Stop after the triples

  10. How many derangements of 4 items are there? Give a number.

    Answer: ______________

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Answer key

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

Count the union without double counting W1-mt_MgU-IMpVHA-s1

  1. 45 · Add both clubs then subtract the overlap counted twice: 30 plus 25 minus 10 = 45.
  2. 45 · Add both clubs, then subtract the overlap counted twice: 30 plus 25 minus 10 is 45.
  3. 31 · Singles sum to 47, pairs sum to 19, triple is 3: 47 minus 19 plus 3 is 31.
  4. True · Shared members appear in both singles and once in the overlap, so two counts minus one leaves exactly one.
  5. 2 · Only the cyclic shifts 231 and 312 avoid every fixed point, so there are 2 derangements.
  6. 31 · Singles sum to 47, pairs sum to 19, triple is 3: 47 minus 19 plus 3 = 31.
  7. 9 · Sieving the 4 fixed-point sets gives 24 minus 24 plus 12 minus 4 plus 1, which is 9.
  8. 2 · Only the orders 231 and 312 avoid every fixed point, so there are 2 derangements.
  9. Subtract the quadruple overlaps · The signs keep alternating, so after adding triples the next correction is a subtraction.
  10. 9 · Inclusion-exclusion over the 4 fixed-point sets gives 24 minus 24 plus 12 minus 4 plus 1 = 9.
Worksheet · LightMySky