Counting Without Double-Counting: Inclusion–Exclusion
A club of 30 students is surveyed about activities. Results: 18 play a sport, 12 play an instrument, and 6 do both. A student, Omar, computes the number who do at least one as 18 + 12 = 30 and concludes "everyone does at least one." A second student, Kim, computes the number who do neither as 30 − 30 = 0. The teacher suspects double-counting and wants the class to reason carefully about overlapping sets before trusting the totals.
1. Explain the double-counting error and compute the correct number who do at least one activity.
Read the worked answer
Adding 18 + 12 = 30 counts the 6 students who do both twice-once in the sport group, once in the instrument group. Those 6 are a single person counted in two categories.
Use inclusion–exclusion: |Sport ∪ Instrument| = |Sport| + |Instrument| − |Both| = 18 + 12 − 6 = 24.
So 24 students do at least one activity, not 30. Omar's total coincidentally equals the club size, which masked the error-it made "everyone does something" look plausible. Subtracting the overlap once corrects the double count.
2. Using the corrected figure, find how many do neither, and check consistency with the club total.
Read the worked answer
Students doing neither = total − (at least one) = 30 − 24 = 6.
Kim's answer of 0 was wrong because it rested on Omar's inflated 30; once at-least-one is corrected to 24, the neither-group is 6, not 0.
Consistency check by region:
- Sport only = 18 − 6 = 12
- Instrument only = 12 − 6 = 6
- Both = 6
- Neither = 6
- Sum = 12 + 6 + 6 + 6 = 30 ✓
Every student is counted exactly once, confirming both the at-least-one figure (12 + 6 + 6 = 24) and the neither figure (6). The lesson: when sets overlap, subtract each shared element the right number of times, and verify by summing disjoint regions to the known total.