Overlapping

[Abdulla]

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 is C

[cramya]

Use 3 set formula

Total (unique objects in set) = grp1 + grp2+ grp3 + none - (1 and 2) - (2 and 3) - (1 and 3) - 2 (1 and 2 and 3)


Total (unique objects in set) = grp1 + grp2+ grp3 + none - {(1 and 2 + 2 and 3) + (1 and 3) } - 2 (1 and 2 and 3)

59 = 27 + 22 +28 -6 - 2 (1 and 2 and 3)


2 (1 and 2 and 3) = 27 + 22 +28 -6 - 29


2 (1 and 2 and 3) = 12

(1 and 2 and 3) = 6

A)

[Abdulla]

Use 3 set formula

Total (unique objects in set) = grp1 + grp2+ grp3 + none - (1 and 2) - (2 and 3) - (1 and 3) - 2 (1 and 2 and 3)


Total (unique objects in set) = grp1 + grp2+ grp3 + none - {(1 and 2 + 2 and 3) + (1 and 3) } - 2 (1 and 2 and 3)

59 = 27 + 22 +28 -6 - 2 (1 and 2 and 3)


2 (1 and 2 and 3) = 27 + 22 +28 -6 - 29


2 (1 and 2 and 3) = 12

(1 and 2 and 3) = 6

A)

You're right 6 is C, however could you explain more about the second part of the formula which is
Total (unique objects in set) = grp1 + grp2+ grp3 + none - (1 and 2) - (2 and 3) - (1 and 3) - 2 (1 and 2 and 3)

[cramya]

grp1 and grp2 -> Objects in grp 1 and grp 2

grp1 and grp2 and grp3-<> Objects in grp 1 and grp 2 and grp3

(grp1 and grp2) - (grp2 and grp3) - (grp1 and grp3) - 2 (grp1 and grp2 and grp3)

Since u r double counting this adjustment is part of the formula.

All I did was change (grp1 and grp2) - (grp2 and grp3) - (grp1 and grp3)

to -( (grp1 and grp2) +(grp2 and grp3) +(grp1 and grp3))

(grp1 and grp2) + (grp2 and grp3) + (grp1 and grp3) is given to be 6 in the problem.

Hope this helps!

[Abdulla]

grp1 and grp2 -> Objects in grp 1 and grp 2

grp1 and grp2 and grp3-<> Objects in grp 1 and grp 2 and grp3

(grp1 and grp2) - (grp2 and grp3) - (grp1 and grp3) - 2 (grp1 and grp2 and grp3)

Since u r double counting this adjustment is part of the formula.

All I did was change (grp1 and grp2) - (grp2 and grp3) - (grp1 and grp3)

to -( (grp1 and grp2) +(grp2 and grp3) +(grp1 and grp3))

(grp1 and grp2) + (grp2 and grp3) + (grp1 and grp3) is given to be 6 in the problem.

Hope this helps!

Thanks Cramya, one more thing.. why did you multiplied the bolded formula by 2 ?