Question

In a competition, a school awarded medals in different categories. 36 medals in dance, 12 medals in dramatics and 18 medals 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?

• A. 7
• B. 5
• C. 13
• D. 2

Hint:

For three-set medal problems, first compute $S_2=\sum$ of pairwise intersections using inclusion-exclusion. Then subtract $3\times$ the "all three" count to get the number in exactly two categories.

Answer 1

The Correct Option is C

Step 1: Let \(p,q,r\) be the number of people in exactly one, exactly two, and all three categories. Given \(r=4\) and total \(p+q+r=45\), so \(p+q=41\).

Step 2: Total medals \(36+12+18=66=p+2q+3r\), so \(p+2q=54\).

Step 3: Subtract: \(q=54-41=13\). \[ \boxed{13} \]