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MATH SETS AND COUNTING

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There are a total of 103 foreign language students in a high school where they offer only Spanish, French, and German.

You are given the following details for this semester:

44 take Spanish.

44 take French.

42 take German.

7 take Spanish and French but not German.

5 take Spanish and German but not French.

5 take French and German but not Spanish.

22 students are taking at least two languages.

(1) How many students take all three languages in this semester?

(2) How many students take only French in this semester?

  • MATH SETS AND COUNTING -

    There are a total of 103 foreign language students in a high school where they offer only Spanish, French, and German.
    Call these groups S F G SF SG FG FGS

    You are given the following details for this semester:
    7 take Spanish and French but not German.
    SF = 7
    5 take Spanish and German but not French.
    SG = 5
    5 take French and German but not Spanish.
    GF = 5

    44 take Spanish.
    S + SF + SG + FGS = 44 -> S + FGS = 32
    44 take French.
    F + SF + FG + FGS = 44 -> F + FGS = 32
    42 take German.
    G + SG + FG + FGS = 42 -> G + FGS = 32
    22 students are taking at least two languages.
    SF + SG + GF + FGS = 22 -> FGS = 22 - 7- 5 -5 = 5
    (1) How many students take all three languages in this semester? FGS = 5

    (2) How many students take only French in this semester?
    F + FGS = 32 -> F = 32 - 5 = 17

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