Problem Solving

Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?


A 2

B 5

C 6

D 8

E 9

OA: C


I really don't get this one. I used Venn diagrams but I keep getting as a result 2.

I used the following approach:
Since there are 59 students total and 6 are in both clubs we need to substract those students. In a Venn diagram we work from the inside out so:

P=22
H=27
W=28

Here we have a total of 77 students. So we need to substract those that are in two and in three clubs.

So let's say there are 6 students in both poetry and writing club, and lets put X as the number of students in all 3 clubs, we would then have:

(22P-6-x)+(27H-X)+(28W-6-X)=

-3X+65=59
-3X=-6

X=2

I'm doing something wrong, but I don't get it where is the mistake. Am I assuming wrong about the people in both clubs? Should I substract a 6 one more time?