Discussion
Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?
Answer: C
4 students watched all three films.
Let x students watch all three; the other \(40 - x\) watched exactly one film each. Counting viewings film by film, each single-film student is counted once and each all-three student three times:
\[ \begin{aligned} 13 + 16 + 19 &= (40 - x) + 3x \\ 48 &= 40 + 2x \Rightarrow x = 4 \end{aligned} \]
Check: 36 students saw one film and 4 saw all three, giving \(36 + 12 = 48\) viewings. With 0 or 2 students watching all three, the viewings would total only 40 or 44; with 8 they would total 56.
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