Question

In a group of 40 students, 25 applied to Columbia and 30 applied to Cornell. If 3 students applied to neither Columbia nor Cornell, how many students applied to both schools?

Answer 1

Let A = # students who applied ONLY to Columbia, B = # students who applied ONLY to Cornell.

Since 3 applied to neither school, there are 37 who applied to either Columbia, Cornell or both.
If 30 of them applied to Cornell, there are 7 that didn't apply to Cornell. Hence they only applied to Columbia. That means that of the 25 Columbia applicants, 25 - 7 = 18 must have applied to both Columbia and Cornell.
So your answer is "18 students applied to both schools."