lheiannie07 wrote:In a group of 20 people, 5 of them belong to the golf club, 7 to the swim club, and 9 to the tennis club. If 2 of the people belong to all three clubs and 3 belong to exactly two of the three clubs, then how many of 20 people belong to neither of the three clubs?

A. 1
B. 2
C. 4
D. 6
E. 11

Here is a useful formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC) + NONE

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
T = 20.
Golf = 5.
Swim = 7.
Tennis = 9.
Exactly 2 of the groups = 3.
All 3 groups = 2.
Let N = none.

Plugging these values into the formula, we get:
20 = 5 + 7 + 9 - 3 - 2(2) + N
20 = 14 + N
6 = N.

The correct answer is D.