# How many students like all the three subjects?

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?

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There is presumably a typo. The usual meaning of "$20$ like English" in questions of this type is that $20$ like English and maybe some other things. Then it is impossible for $24$ to like Math and English. – André Nicolas Jan 20 '13 at 2:23

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 $$|M\cap S\cap E|=21.$$
As an aside, I do agree with Andre Nicolas; if $20$ like English, how can $24$ like Math and English? However, whatever the typo is, this would be the general method to solution. – Clayton Jan 20 '13 at 2:50