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math

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in a school of 100 students, 59 are taking math, 51 are in english, 35 are in history, 24 are in both math and english, 19 are in both english and history, 13 in both math and history, and 11 in all three courses. how many students are taking:
a) none of the classes?
b) only math?
c) history and english?
d) only english?
e) exactly two of the three classes?

  • math - ,

    I drew three overlapping circles.

    all three =11

    math and history only 13-11 = 2
    english and history only 19-11 = 8
    math and english only 24-11 = 13

    so math only = 59-13-11-2 = 33
    english only = 51-13-11-8 = 19
    history only = 35-2-11-8 = 14

    add them all up and you get 100, so no students are taking none.

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