SOLUTION: A group of 2193 students were surveyed about the courses they were taking at their college with the following results: 1192 students said they were taking Dance. 1143 students

Venn diagram clues are nearly always give in the reverse order
from the order in which you use them, so we take them in the
reverse order and begin with the last clue and end with the first
clue:

So V=372. We fill that in:

So IV+V=646, so substituting V=372, IV=274. We fill that in:

So V+VI=646, so substituting V=372, VI=325. Fill that in:

So II+V=583, so substituting V=372, we get II=211. We fill that in:

So IV+V+VI+VII=n(M)=1199 and substituting for IV,V and VI gives VII=228.
We fill that in:

So II+III+V+VI=n(E)=1199 and substituting for II,V and VI gives III=235.
We fill that in:

So I+II+IV+V=n(D)=1199 and substituting for II,IV and V gives I=335.
So fill that in:

So I+II+III+IV+V+VI+VII+VIII=n(U)=n(universal set)=2193, and substituting for
I,II,III,IV,V,VI and VII gives VIII=213.  So fill that in (the number who
took none of the courses):

They were I,IV and VII, or 335+274+228=837

That's all 2193 except those 213 in VIII who took none of the courses.

2193-VIII = 2193-213 = 1980

Those that took Dance & English were II+V = 211+372
Those that took English & Math were V+VI = 372+325
But WE MUST BE CAREFUL HERE!!! If we added all those together we
would be counting the 372 TWICE! So we must add it only once, so the
answer here is II+V+VI = 211+372+325 = 908

 

That's VI, which is 325 

 
Those that took Dance were given as 1192 which was I+II+IV+V
Those that didn't take English were I+IV+VII+VIII.
But AGAIN WE MUST BE CAREFUL HERE!!! If we added all those together we
would be counting the I and IV TWICE! So we must add those only once, so the
answer here is I+II+IV+V+VII+VIII = 335+211+274+372+228+213 = 1633    

We've already answered that.  It's VIII or 213.

Edwin