75 students were interviewed. 39 took Mathematics, 32 took Biology, while 12 took none of these subjects. How many students took both Mathematics and Biology?
8
Use the two-set inclusion-exclusion formula. If M is the set of students who took Mathematics and B the set who took Biology, then \(n(M \cup B) = n(M) + n(B) - n(M \cap B)\).
Step 1 — Total interviewed \(= 75\). Students who took at least one of the two subjects \(= 75 - 12 = 63\), since 12 took neither.
Step 2 — Substitute in the formula: \(63 = 39 + 32 - n(M \cap B)\).
Step 3 — Solve: \(n(M \cap B) = 39 + 32 - 63 = 8\).
Hence, 8.
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