# I am studying for the p exam and realized I really need to brush up on my basic set theory, and am having trouble with this question. In a survey on Popsicle flavor preferences of kids aged 3-5, it was found that: • 22 like strawberry.• 25 like blueberry.• 39 like grape.• 9 like blueberry and strawberry.• 17 like strawberry and grape.• 20 like blueberry and grape.• 6 like all flavors.• 4 like none. Apparently the answer is 50, but I can't seem to figure out how to arrive at this

I am studying for the p exam and realized I really need to brush up on my basic set theory, and am having trouble with this question.
In a survey on Popsicle flavor preferences of kids aged 3-5, it was found that:
- 22 like strawberry.
- 25 like blueberry.
- 39 like grape.
- 9 like blueberry and strawberry.
- 17 like strawberry and grape.
- 20 like blueberry and grape.
- 6 like all flavors.
- 4 like none.
Apparently the answer is 50, but I can't seem to figure out how to arrive at this
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Note that
$P\left(A\cup B\cup C\right)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right).$
Now let A be "kid likes strawberry", B be "kid likes blueberry", C be "kid likes grape", and n be the number of kids.
P(A)=22/n, P(B)=25/n, P(C)=39/n, with intersections 9/n, 17/n, 20/n, 6/n.
Then the probability of liking any flavor at all is
$P\left(A\cup B\cup C\right)=\left(22+25+39-9-17-20+6\right)/n=46/n.$
Also, the probability of not liking any flavor at all is 4/n, but also
$4/n=P\left({A}^{c}\cap {B}^{c}\cap {C}^{c}\right)=1-P\left(A\cup B\cup C\right)=1-46/n.$
Therefore,
$\frac{4}{n}=1-\frac{46}{n}=\frac{n-46}{n},$
so 4=n−46, and so n=50.