Question

In a class of 58 students, 20 follow cricket, 38 follow hockey and 15 follow basketball. Three students follow all the three games. How many students follow exactly two of these three games?

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Answer 1

Number of students who follow cricket n(A) = 20

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

Number of students who follow basketball n(c) = 15

Number of students follow all three games (A $\cap$ B $\cap$ C) = 3

From Venn diagram we know,

n(A $\cup$ B $\cup$ C) = ∑ (A) - ∑ n(A $\cap$ B) + n(A $\cap$ B $\cap$ C)

58 = n(A) + n(B) + n(C) - ∑ n(A $\cap$ B) + 3

58 = 20 + 38 + 15 - ∑n(A $\cap$ B) + 3

∑n(A $\cap$ B) = 76 - 58 = 18

n(A $\cap$ B) + n(B $\cap$ C) + n(C $\cap$ A) = 18

So, number of students follows exactly two of three games

∑n(A $\cap$ B) = 18