Inclusion-Exclusion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Count the union without double counting

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

  1. For three sets A, B and C, 6 members are in A and B only, and 2 are in all three. What size of A ∩ B belongs in the inclusion-exclusion formula?

    • 8
    • 6
    • 4
    • 2

  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 and 12 like juice. 8 like tea and coffee, 6 like tea and juice, and 5 like coffee and juice. These pair counts include the 3 who 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. Five distinct letters go one per matching envelope. F1 contains placements with letter 1 in its own envelope; F2 does the same for letter 2. How many placements are in F1 ∩ F2? Other letters may also reach their own envelopes.

    • 24
    • 2
    • 6
    • 0

  6. Four sets have sizes summing to 70, six pair-intersection sizes summing to 40, four triple-intersection sizes summing to 16, and 3 members in all four. Intersection counts include members in further sets. How many elements are in the union?

    Answer: ______________

  7. Four distinct items numbered 1 to 4 are placed one per position numbered 1 to 4. How many orders have no item in its matching position? Give a number.

    Answer: ______________

  8. Three distinct letters numbered 1, 2 and 3 are placed in matching envelopes numbered 1, 2 and 3, one letter per envelope. In how many placements does no letter go into its matching envelope?

    • 2
    • 1
    • 6

  9. For four sets, an element in all four contributes 4 - 6 + 4 = 2 after adding singles, subtracting pairs and adding triples. Which final correction makes its contribution 1?

    • Add 1 for the intersection of all four sets again
    • Subtract 1 for the intersection of all four sets
    • Subtract 2 for the intersection of all four sets

  10. Each of four labelled tasks is assigned to exactly one of three named people. Each person must receive at least one task. How many assignments meet this requirement?

    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. 8 · A ∩ B includes members also in C. Add the 6 in A and B only to the 2 in all three: 6 + 2 = 8.
  2. 45 · Add the two club sizes, then subtract the overlap that was 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. 6 · Fix letters 1 and 2. The three remaining letters can occupy the remaining envelopes in 3! = 6 ways. Further fixed points are allowed.
  6. 43 · Use singles minus pairs plus triples minus the intersection of all four: 70 - 40 + 16 - 3 = 43.
  7. 9 · There are 24 orders. The intersection sums for fixing specified singles, pairs, triples and all four positions are 24, 12, 4 and 1. Thus 24 - 24 + 12 - 4 + 1 = 9.
  8. 2 · Only the orders 231 and 312 avoid every fixed point, so there are 2 derangements.
  9. Subtract 1 for the intersection of all four sets · The element appears in the intersection of all four sets. Subtracting that term removes one copy, changing its contribution from 2 to 1.
  10. 36 · Start with 3^4 = 81 assignments. Subtract 3 × 2^4 = 48 for missing a specified person, then add 3 × 1^4 = 3 for missing a specified pair. The result is 36.
Worksheet · LightMySky