In this problem, we have a universal set W of all 970 students.


We also have three subsets: B (banning) of 390 students;

                            D (dance)   of 930 students,

                   and      A (against both) of 20 students.


Notice that the sub-sets B and D may have non-empty intersection; while the sub-set A is DISJOINT from both B and D.


THEREFORE, the union of B and D has  970 - 20 = 950 students.


So, we can write  n(B U D) = n(B) + n(D) - n(B & D),  or

                     950   = 390  + 930 -  n(B & D).


From this equation, you get the ANSWER to the problem's question  n(B & D) = 390 + 930 - 950 = 370.


ANSWER.  370 students voted in favor of both proposals.