# Counting Venn Diagram problem

"In a survey of 185 university students, 91 were taking a history course, 75 were taking a biology course, and 37 were taking both. How many were taking a course in exactly one of these subjects?"

I know the answer is 92 from the back of the book. I know also that when I draw out a Venn diagram I find that 37 take both, 54 take History, 38 take Biology. In my text book it says that $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ yet, using this does not work. Can someone please explain why?

You got the right answer - $38+54=92$. The equation on the textbook is NOT appropriate for this problem. That equation is for finding how many unique people there are, not how many take ONLY 1 class. For this problem, the appropriate equation is:

$$n(A \cup B) = n(A) + n(B) - 2n(A \cap B)$$

And more specific to the problem, one will be $H$ (History) and the other will be $B$ (Biology):

$$n(H \cup B) = n(H) + n(B) - 2n(H \cap B)$$

What do we know? We know that $H \cap B$, which is History AND Biology is 37. Then we know that $H$ is 91, and $B$ is 75. Thus:

$$n(H \cup B) = 91 + 75 - 2(37)=92$$

• It's kind of confusing how you make no notational difference between sets and their cardinality. – ruler501 Mar 21 '14 at 21:25
• @ruler501 Fixed. – Shahar Mar 21 '14 at 21:28

The reason the formula doesn't work is that you are calculating the number of people taking 1 and/or the other $A \cup B$. You want the ones taking just one of them. To get that you need $n(A \cup B) - n(A \cap B) = n(A)+n(B)-2n(A \cap B)$