MHB Counting Problem: In a school 315 girls play at least one sports

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In a school, 315 girls participate in at least one sport, with 100 in fall, 150 in winter, and 200 in spring sports. A Venn diagram is suggested to visualize the relationships among the groups. The formula for calculating the total number of participants across overlapping sets is provided, which includes intersections of the sets. It is noted that while specific intersections are not given, the total for girls playing exactly two sports is known to be 75. The goal is to determine how many girls play all three sports.
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In a school 315 girls play at least one sport. 100 play a fall sport, 150 play a winter sport, and 200 play a spring sport. If 75 girls play exactly 2 sports, how many play three?
 
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Hello, and welcome to MHB! (Wave)

I would begin by constructing a Venn diagram:

View attachment 9104

We've got 7 variables...can you construct equations involving these variables from the given information?
 

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Hi Avro.

You can also use this formula for any sets $A$, $B$, $C$:
$$|A\cup B\cup C|\ =\ |A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|.$$
So, in this problem, $A$ might be the set of girls playing fall sports, $B$ the set of those playing winter sports, and $C$ the set of those playing spring sports; then you want to find $|A\cap B\cap C|$. Also, note that while you are not given $|A\cap B|$, $|B\cap C|$ or $|C\cap A|$ separately, you are given $|A\cap B|+|B\cap C|+|C\cap A|$, which you can use in the formula above
 
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