How many students like all the three subjects?

Question

Among 64 students, 28 of them like Science, 41 like Mathematics and 20 like English. 24 of them like both Math and English. 12 students like both Science and English. 10 students like both science and math. How many students like all the three subjects?

Answer 1

Using the fact that $|A\cup B|=|A|+|B|-|A\cap B|$, we can extend this to a notion of three sets, namely the students that like math (M), science (S), and english (E). Then we have $$64=|M\cup S\cup E|=|M|+|S|+|E|-|M\cap S|-|M\cap E|-|S\cap E|+|M\cap S\cap E|.$$ Substituting in and solving for the last term, we have $$|M\cap S\cap E|=21.$$