SOLUTION: There are 64 students in a class with the students offering at least one of the subjects literature, accounts and economics: 35 students offer literature, 32 offers accounts, and 4

    There are 64 students in a class with the students taking at least one of the subjects literature, accounting and economics: 
    35 students take literature, 32 take accounting, and 40 take economics. 17 take both literature and accounting, 
    15 take literature and economics, 12 take economics and accounting. How many students take two subjects only?

The idea of the solution is very simple:

    To determine how much students study only literature and accounting, we take the number of 17 of those who study 
    literature and accounting and subtract from it the number of those who study all three subjects.


    Similarly, to determine how much students study only literature and economics, we take the number of 15 of those 
    who study literature and economics and subtract from it the number of those who study all three subjects.


    Finally, to determine how much students study only economics and accounting, we take the number of 12 of those 
    who study economics and accounting and subtract from it the number of those who study all three subjects.


So, we would get the answer momentarily, had we know the number of those who study all three subject.


Thus, it is the major unknown in this problem.


To find it, use this remarkable formula of the elementary set theory:

   n(L U A U E) = n(L) + n(A) + n(E) - n(LnA) - n(LnE) - n(AnE) + n(LnAnE)    (1)

which is valid for any three subsets L, A and E of the universal set U.


In formula (1),  LnA  is the intersection of the subsets L and A;
                 LnE  is the intersection of the subsets L and E; 
                 AnE  is the intersection of the subsets A and E;   and
                 LnAnE is the intersection of the THREE subsets L, A and E.

n(X) symbolizes the number of elements in subset X.


In our case L is the set of those who study literature;  A is the set of those who study accounting;  
LnA  is the intersection of sets L and A, i.e. the set of those who study literature and accounting; and so on  
(all designations are self-explanatory).

The last notice before applying the formula (1) is that the union  L U A U E  is the entire class consisting of 64 students,
so n(L U A U E) = 64.


Now substitute all other given input data into the formula (1). You will get


    64 = 35 + 32 + 40 - 17 - 15 - 12 + n(LnAnE).


in the last formula, all the terms are known except the last one, so we easily get


    n(LnAnE) = 64 - 35 - 32 - 40 + 17 + 15 + 12 = 1.


Now we are in position to answer each question:

    - the number of those who study L and A is  n(LnA) - n(LnAnE) = 17-1 = 16.
    - the number of those who study L and E is  n(LnE) - n(LnAnE) = 15-1 = 14.
    - the number of those who study A and E is  n(AnE) - n(LnAnE) = 12-1 = 11.


And, finally, the number of those who study exactly TWO subjects is  16 + 14 + 11 = 31.    ANSWER


ANSWER.  The number of those who study exactly TWO subjects is  31.