200 registered, but only 190 registered for S, D or R. The remaining 10 are registered for none of these.
Define the following variables a-g, where each variable represents the numbers registered:
a) S only
b) R only
c) D only
d) S+R only
e) S+D only
f) R+D only
g) all three
Registered for S: a+d+e+g=107
Registered for R: b+d+f+g=90
Registered for D: c+e+f+g=63
g=15
There is an ambiguity: is d=35 or is d+g=35 so d=20? similarly e=23 or 8.
This causes an ambiguity in d+e+g=73 or 43 and a=34 or 64.
Also, b+f=40 or 55 and c+f=25 or 40.
a+b+c+d+e+f+g=190, a+b+c+d+e+f=175.
If we add together the three equations for the registered students we get:
a+b+c+2d+2e+2f+3g=260, a+b+c+2d+2e+2f=215.
Subtracting the two equations containing a+b+c we get:
d+e+f=40=d+e+x, because x by definition is the same as f.
We know that d+e=58 or 28.
Clearly we must choose 28 because 58 exceeds 40. Therefore x=40-28=12, so d=20 and e=8,a=64.
Solution: x=12. If D+R is meant to include those students who registered for all three (like S+R and S+D) then x=27.