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

- Manhattan GMAT CAT

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

Here is the formula for triple-overlapping groups A, B and C:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to subtract the overlaps.
When we count the total number of elements in TWO of the groups (AB + BC + AC), these elements get counted TWICE.
So that we don't double-count these elements, we subtract them from the total ONCE.
When we count the total number of elements in all THREE groups (ABC), these elements get counted THREE TIMES.
So that we don't triple-count these elements, we subtract them from the total TWICE.

Let x = the number of students in all clubs.
Then according to the formula above:
59 = 22 + 27 + 28 - 6 - 2x
59 = 71 - 2x
x = 6.

The correct answer is C.

See Image

Hi
IMO the answer is C

T = P + H + W - (PH + HW + WP) - 2(PHW)

Then according to the formula above:
59 = 22 + 27 + 28 - 6 - 2x
59 = 71 - 2x
x = 6

thanks and regards