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21 December, 08:46

In a group of Explore students, 38 enjoy video games, 12 enjoy going to the movies and 24 enjoy solving mathematical problems. Of these, 8 students like all three activities, while 30 like only one of them.

How many students like only two of the three activities?

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  1. 21 December, 09:10
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    The number of students that like only two of the activities are 34

    Step-by-step explanation:

    Number of students that enjoy video games, A = 38

    Number of students that enjoy going to the movies, B = 12

    Number of students that enjoy solving mathematical problems, C = 24

    A∩B∩C = 8

    Here we have;

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

    = 38 + 12 + 24 - n (A∩B) - n (B∩C) - n (A∩C) + 8

    Also the number of student that like only one activity is found from the following equation;

    n (A) - n (A∩B) - n (A∩C) + n (A∩B∩C) + n (B) - n (A∩B) - n (B∩C) + n (A∩B∩C) + n (C) - n (C∩B) - n (A∩C) + n (A∩B∩C) = 30

    n (A) + n (B) + n (C) - 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) + 3·n (A∩B∩C) = 30

    38 + 12 + 24 - 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) + 24 = 30

    - 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) = - 68

    n (A∩B) + n (B∩C) + n (A∩C) = 34

    Therefore, the number of students that like only two of the activities = 34.
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