Question

A college awarded 38 medals in volleyball, 15 in football, and 20 in basketball. The medals were 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?

Answer 1

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.