Discrete Math

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A professor gave his 40 students a test with three questions. Every student answered at least one question. Ten didn't answer the first question. 14 didn't answer the second question. 12 didn't answer the third question. If 18 students answered all three questions, how many answered exactly one question?

Thanks for your help

  • Discrete Math -

    t = two answers , x = one answer

    (3 * 40) - (3 * 18) - 10 - 14 - 12 = 2 t + x

    40 - 18 = t + x

    solve the system for x

  • Discrete Math -

    Another way to solve.
    Let
    A=# of students who answered the first question
    B=# of students who answered the second question
    C=# of students who answered the third question

    Then
    we are given
    B+C-|B∩C| = 10
    A+C-|A∩C| = 14
    A+B-|A∩B| = 12
    Add three equations
    2A+2B+2C-(|A∩B|+|B∩C|+|C∩A|)=36....(1)


    But the total number who answered one OR two question is 40-18=22, or
    A+B+C-|A∩B|- |B∩C|-|C∩A|=22.....(2)

    (1)-(2)
    A+B+C=36-22=14

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