The answer is not 37, even though the sum of the numbers above is 37. For example, while 12 students failed Algebra, 2 of those students also failed Biology, 6 also failed Chemistry, and 1 of those failed all three subjects. In fact, that 1 student who failed all three subjects is counted a total of 7 times in the total 37. To clarify things, let us think of the students who failed Algebra as the elements of the set \(A\text{,}\) and similarly for sets \(B\) and \(C\text{.}\) The one student who failed all three subjects is the lone element of the set \(A \cap B \cap C\text{.}\) Thus, in Venn diagrams:
in-context