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Math Help - simple set example

  1. #1
    Mar 2013

    Question simple set example

    I can't figure out why the correct anwer is a)

    Each of 35 students learns at least one of these languages: English, German, French.
    5 students study all of these languages.
    20 students study English.
    15 students study German
    10 students study French.

    Decide which of these statement is true:

    a) No student exists, who learns exactly 2 languages.
    b) At least one student learns exactly 2 languages.
    c) At least one student learns French and German and doesn't learn English.
    d) Described situation cannot happen.
    d) None of the statements is correct.

    I would greatly appreciate if you describe how you got the result.
    Thank you.
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  2. #2
    MHF Contributor
    Oct 2009

    Re: simple set example

    Let x, y and z denote (in some order) the number of students who study exactly two languages. Then by the inclusion–exclusion principle, we have

    20 + 15 + 10 - (x + 5) - (y + 5) - (z + 5) + 5 = 35

    from where x + y + z = 0.
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