Discussion
In a competition, a school awarded medals in different categories. 36 medals in dance, 12 in dramatics and 18 in music. If these medals went to a total of 45 persons and only 4 persons got medals in all the three categories, how many received medals in exactly two of these categories?
Answer: C
Correct answer: 13
Write the inclusion–exclusion identity for three sets, with \(D\), \(R\) and \(M\) for dance, dramatics and music.
\[n(D \cup R \cup M) = n(D)+n(R)+n(M) - \Sigma\,\text{pairs} + n(D \cap R \cap M)\]
\[45 = 36+12+18 - \Sigma\,\text{pairs} + 4 = 70 - \Sigma\,\text{pairs}\]
So the pairwise intersections total \(\Sigma\,\text{pairs} = 25\).
Each person holding all three medals is counted in all three of those pairwise intersections, so subtract them three times over to leave those in exactly two:
\[25 - 3 \times 4 = 25 - 12 = 13\]
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