The Overlapping Survey
A school surveys 240 students about three optional activities: chess club (C), drama club (D), and debating club (B). The results show that 130 joined C, 110 joined D, and 90 joined B. Exactly 40 students joined both C and D, exactly 30 joined both D and B, and exactly 25 joined both C and B. The number of students who joined all three clubs is unknown and is denoted by x. The survey also reports that 35 students joined none of the three clubs. A teacher claims these figures are consistent and asks the class to determine x, then to check whether the resulting counts are all possible.
1. Use the inclusion–exclusion principle to write an equation for x and solve it.
Read the worked answer
Let |C∪D∪B| = 240 − 35 = 205. By inclusion–exclusion,
|C∪D∪B| = 130 + 110 + 90 − 40 − 30 − 25 + x = 235 + x.
So 235 + x = 205, giving x = −30.
A negative triple intersection is impossible, so the reported figures are inconsistent. The teacher's claim fails.
2. Suppose the number of students in none of the clubs is changed to a value n. Find the range of n for which a valid non-negative integer x exists, and interpret the boundary values.
Read the worked answer
We need x = (240 − n) − 235 = 5 − n, with 0 ≤ x ≤ 25 (x cannot exceed the smallest pairwise intersection, 25).
So 0 ≤ 5 − n ≤ 25 gives −20 ≤ n ≤ 5.
Since n is a count, 0 ≤ n ≤ 5.
- n = 5 gives x = 0: no student is in all three clubs.
- n = 0 gives x = 5: five students are in all three, and every student is in at least one club.
For n > 5, x would be negative; for n < 0, impossible as a count.