In a class of 75 students, 40 follow cricket, 38 follow hockey and 10 follow basketball. Five students follow all the three games. Then the number of students who follow exactly two of these three games are ___.

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Solution

Number of students who follow cricket n(A) = 40

Number of students who follow hockey n(B) = 38

Number of students who follow basketball n(C) = 10

Number of students follow all three games (A∩B∩C) = 5

From Venn diagram we know,

n(A∪B∪C) = ∑ (A) - ∑ n(A∩B) + n(A∩B∩C)

75 = n(A) + n(B) + n(C) - ∑ n(A∩B) + 5

75 = 40 + 38 + 10 - ∑n(A∩B) + 5

∑n(A∩B) = 93 - 75 = 18

n(A∩B) + n(B∩C) + n(C∩A) = 18

So, number of students following exactly two of three games

∑n(A∩B) = 18

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