The Overlapping Membership Ledger
A district sports council has 240 registered members. Each member must hold at least one of three passes: Swimming (S), Athletics (A), or Rowing (R). The registrar reports that 130 hold S, 110 hold A, and 90 hold R. She also reports that 45 members hold both S and A, 30 hold both A and R, and 25 hold both S and R. A grant application requires the council to state how many members hold exactly one pass and how many hold all three. A junior clerk, in a hurry, simply adds the three single-pass totals (130 + 110 + 90 = 330), subtracts the three pairwise totals (45 + 30 + 25 = 100), and claims 230 members hold at least one pass - but the register clearly has 240. He cannot explain the shortfall and asks you to audit the numbers before the grant form is signed.
1. Using inclusion–exclusion, find the number of members holding all three passes, and check whether the registrar's figures are internally consistent with 240 members.
Read the worked answer
Let x be the number holding all three.
By inclusion–exclusion:
|S ∪ A ∪ R| = (130 + 110 + 90) − (45 + 30 + 25) + x = 330 − 100 + x = 230 + x.
Since every member holds at least one pass, |S ∪ A ∪ R| = 240, so 230 + x = 240 and x = 10.
The clerk's error was omitting the +x term: subtracting the pairwise overlaps removes each triple-holder three times too many, so the triple count must be added back once. With x = 10 the figures are consistent.
Check: each pairwise figure (45, 30, 25) must be at least x = 10 - true. Also each single total must be at least the sum of its two pairwise overlaps minus x: for S, 45 + 25 − 10 = 60 ≤ 130 - true. The data are feasible.
2. Determine how many members hold exactly one pass, and how many hold exactly two passes.
Read the worked answer
Use the standard region counts. For a single set, e.g. S only:
|S only| = |S| − |S∩A| − |S∩R| + |S∩A∩R| = 130 − 45 − 25 + 10 = 70.
Similarly:
- A only = 110 − 45 − 30 + 10 = 45
- R only = 90 − 30 − 25 + 10 = 45
Exactly one pass: 70 + 45 + 45 = 160.
Exactly two passes (pairwise overlaps minus the triple, summed):
- S∩A only = 45 − 10 = 35
- A∩R only = 30 − 10 = 20
- S∩R only = 25 − 10 = 15
Total exactly two = 35 + 20 + 15 = 70.
Consistency check: 160 + 70 + 10 = 240. ✓ The grant form should state 160 holders of exactly one pass and 10 holders of all three.