Discussion
In a competition, 36 medals were awarded 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
Let D, R, and M be the sets of people who got medals in Dance, Dramatics, and Music respectively.
Given: n(D)=36, n(R)=12, n(M)=18, n(D ∪ R ∪ M)=45, n(D ∩ R ∩ M)=4.
We use the formula:
n(D∪R∪M) = n(D)+n(R)+n(M) - [n(D∩R) + n(R∩M) + n(M∩D)] + n(D∩R∩M)
45 = (36 + 12 + 18) - [Sum of intersections of pairs] + 4
45 = 66 - [Sum of pairs] + 4
45 = 70 - [Sum of pairs]
Sum of pairs = n(D∩R) + n(R∩M) + n(M∩D) = 25.
The number of people who received medals in exactly two categories is given by:
[n(D∩R) + n(R∩M) + n(M∩D)] - 3 × n(D∩R∩M)
= 25 - 3 × 4 = 25 - 12 = 13.
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