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Question
A college awarded 38 medals in volleyball, 15 in football, and 20 in basketball. The medals awarded to a total of 58 players and only 3 players got medals in all three sports. How many received medals in exactly two of the three sports?
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Solution
Let A = Set of students who received medals in volleyball.
B = Set of students who received medals in football.
C = Set of students who received medals in basketball.
n(A) = 38, n(B) = 15, n (C) = 20, n(A ∪ B ∪ C) = 58, n(A ∩ B ∩ C) = 3.
n(A ∪ B ∪ C)
= n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)
∴ 58 = 38 + 15 + 20 − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + 3
∴ n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = 18 .......(i)
Number of players who got exactly two medals = p + q + r
Here, s = n(A ∩ B ∩ C) = 3
n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = 18 …[From (i)]
∴ p + s + s + r + q + s = 18
∴ p + q + r + 3s = 18
∴ p + q + r + 3(3) = 18
∴ p + q + r = 18 − 9 = 9
∴ Number of players who received exactly two medals = 9.
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