3 set overlapping problem

guerrero wrote:There are three country clubs in town: Abacus, Bradley, and Claymore. Abacus has 300 members, Bradley 400, and Claymore has 450. 30 people belong to both Abacus and Bradley, 40 to both Abacus and Claymore, and 50 to both Bradley and Claymore. 20 people are members of all three clubs. How many people belong to at least 1 country club in town?

I am trying to use the formula-
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Its giving me a wrong answer ..Where am I missing ?

I could solve it via Venn Diag , but it took me ~ 3 mins to solve .
thanks !!

1050

Here is the formula for 3 overlapping groups:

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

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 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.
A = 300.
B = 400.
C = 450.
ABC = 20.

Since 30 belong to A and B, and 20 of these people belong to ALL 3 GROUPS (ABC), the number who belong to A and B but not C = 30-20 = 10.
Thus, AB = 10.
Since 40 belong to A and C, and 20 of these people belong to ALL 3 GROUPS (ABC), the number who belong to A and C but not B = 40-20 = 20.
Thus, AC = 20.
Since 50 belong to B and C, and 20 of these people belong to ALL 3 GROUPS (ABC), the number who belong to B and C but not A = 50-20 = 30.
Thus, BC = 30.

Plugging these values into the formula above, we get:

T = A + B + C - (AB + AC + BC) - 2(ABC) = 300 + 400 + 450 - (10+20+30) - 2(20) = 1050.