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

#### Avro

##### New member
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?

#### MarkFL

Staff member
Hello, and welcome to MHB!

I would begin by constructing a Venn diagram:

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

#### Olinguito

##### Well-known member
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