this problem is easier to solve using venn diagram. Please refer to the attached image.

from the given information

D+A+X+C = P = 22
F+A+X+B = H =27
E+B+X+C = W = 28
(D+F+E) + (A+B+C) + X = 59
A+B+C=6

adding first 3 equations,
(D+E+F)+2(A+B+C)+3X=77
Subtracting the above equation from the 4th equation
(A+B+C)+2X=18
6+2X=18
or X=6 (option c)

Hope this helps.

I am not sure of the Venn Diagram...if someone knows that route...it'll be a good idea to explain...here's what I think and I remember seeing this on one of the forums:

there are total 59 students but according to the question
(22+27+28) signed up which is 18 extra counts. The question states 6 signed up exactly for 2 clubs. we are left with 12 and now out of these the people who have signed in all the 3 clubs count for 2 extra sign ups which is 6 so subtract 6 from 12 and you get 6 who signed up for all three clubs.

just_do_it, your Venn Diagram is not showing. Could you probably re-attach it?

pre-gmat, I was following your explanation but got lost on this bolded part, can you please clearify?

pre-gmat wrote:we are left with 12 and now out of these the people who have signed in all the 3 clubs count for 2 extra sign ups which is 6 so subtract 6 from 12 and you get 6 who signed up for all three clubs.

Thanks!

attaching the file one more time.