# math

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 93 students in the school. There are 35 in the Spanish class, 31 in the French class, and 22 in the German class. There are 12 students that in both Spanish and French, 6 are in both Spanish and German, and 8 are in both French and German. In addition, there are 2 students taking all 3 classes.

If one student is chosen randomly, what is the probability that he or she is not in any of these classes?

If two students are chosen randomly, what is the probability that at least one of them is taking a language class?

Show work or an explanation please!

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1. draw a Venn diagram of three intersecting circles enclosed in a box

label the circles S, F, and G

S/F/G -- 2 ... these are in all the two way intersections
S/F -- 10 + (2)
S/G -- 4 + (2)
F/G -- 6 + (2)
S -- 19 + (2) + (10) + (4)
F -- 13 + (2) + (10) + (6)
G -- 10 + (2) + (4) + (6)

language students
... 19 + 13 + 10 + 6 + 4 + 10 + 2 = 64

non-language ... 93 - 64

"at least one" means that both cannot be non-language

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