KA-0154Inclusion–exclusion with two sets
4 points, difficulty 2 of 3Level 7–8about 105sIn a class of 30 students, 18 play football and 15 play chess. Exactly 7 students do both. How many students play neither?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Adding 18 and 15 counts the 7 who do both twice.
2The strategy
Work out how many play at least one sport first, then take that away from the class.
3The full solution
18 + 15 - 7 = 26 play at least one. 30 - 26 = 4 play neither.
Solution
The reliable way
Adding the two groups counts every student who does both exactly twice, so subtract the overlap once to correct it: 18 + 15 - 7 = 26 students play at least one of the two. Everyone else plays neither, so the answer is 30 - 26 = 4. A useful check is to split the class into four boxes: 11 football only, 8 chess only, 7 both, 4 neither, and 11 + 8 + 7 + 4 = 30. The transferable idea: when two groups overlap, add them and subtract the overlap exactly once.
The elegant way
Football only is 11 and chess only is 8, so 11 + 8 + 7 = 26 are accounted for and 4 are left.
Why this is on the test: Two overlapping groups appear on every paper, and the whole difficulty is remembering to subtract the overlap exactly once.