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There are a total of 152 foreign language students in a high school where they offer only Spanish, French, and German.

You are given the following details for this semester:

58 take Spanish.

71 take French.

66 take German.

9 take Spanish and French but not German.

9 take Spanish and German but not French.

15 take French and German but not Spanish.

38 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?

3. What is the exact probability that two students selected from this school take French? Enter your answer as a rational number:

  • math - ,

    Use Venn diagrams.
    Make 3 overlapping circles, and label them S, F, and G
    Place x in the intersection of all 3
    and then place the 9, 9, and 15 in their correct regions
    In the "Spanish only" part, put in 40-x
    In the "French only" part, put in 47-x
    In the "German only" part, put in 42-x

    We are told 38 are taking at least two languages, that would be the regions showing some overlap, so
    9+9+15+x = 38
    x = 5

    1. 5 take all 3
    2. 42 take only French
    3. prob = (71/152)(70/151) = 2485/11476

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