So in a textbook, the question states:
- There are 90 students and each of them must study at least one of Biology, Physics, or Chemistry. There are 36 students who study Biology, 42 who study Physics, and 40 who study Chemistry. Moreover, 9 study Biology and Physics, 8 study Biology and Chemistry, and 7 study Physics and Chemistry. How many students study all 3 subjects?
Initially, I tried to solve it with the inclusion-exclusion principle. As such, I rearranged the equation $n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)$ into $n(A\cap B\cap C)=n(A\cup B\cup C)-n(A)-n(B)-n(C)+n(A\cap B)+n(A\cap C)+n(B\cap C)$, and substituted the values in, which is $n(A\cap B\cap C)=90-36-42-40+9+8+7$. However, this means that $n(A\cap B\cap C)=-4$, and hence the set has negative cardinality unless I have done something wrong. The answer is the back of the book is $4$, which leads me to believe that there is a mistake in the question.
I also checked with a venn diagram, which also supports the answer being $-4$.
Are there any issues with my working, or is the question incorrect?