thats what i tried to realize:
is it true to say that when u look for ALL or EXACTLY 2 use the 2nd formula and when you look for Neither or Total use the 1st.

ALL of the cofee mixtures sold in a certain store contain either a Colombian, Jamaican or Brazilian cofee or some combinations of these. Of all the mixtures 33 contain Colombian cofee, 43 contain Jamaican cofee and 42 contain Brazilian cofee. Of these, 16 contain atleast Colombian and Jamaican cofees, 18 contain atleast Jamaican and Brazilian cofees, 8 contain atleast Brazilian and colombian cofees and 5 contain all three. How many different mixtures are sold in the store ?
(A) 71
(B) 81
(C) 109
(D) 118
(E) 165

Hello Bunuel -

Quick question on the following Overlapping Set Example -

nifoui wrote:
Can someone confirm whether this formula is true?

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

I am getting confused with all different formulas and have seen people adding the 2*(all three) rather than substracting....


I'd advise to understand the concept rather than to memorize formulas. Look at the diagram below:
Attachment:

Union_3sets.gif [ 11.63 KiB | Viewed 21045 times ]



FIRST FORMULA
We can write the formula counting the total as: .

When we add three groups A, B, and C some sections are counting more than once. For instance: sections 1, 2, and 3 are counted twice and section 4 thrice. Hence we need to subtract sections 1, 2, and 3 ONCE (to get one) and subtract section 4 TWICE (again to count section 4 only once).

Now, in the formula AnB means intersection of A and B (sections 3 and 4 on the diagram). So when we are subtracting (3 an4), (1 and 4), and (2 and 4) from , we are subtracting sections 1, 2, and 3 ONCE BUT section 4 THREE TIMES (and we need to subtract section 4 only twice), therefor we should add only section 4, which is intersection of A, B and C (AnBnC) again. That is how thee formula is derived.

SECOND FORMULA
The second formula you are referring to is: {Sum of Exactly 2 groups members} . This formula is often written incorrectly on forums as Exactly 2 is no the same as intersection of 2 and can not be written as AnB. Members of exactly (only) A and B is section 3 only (without section 4), so members of A and B only means members of A and B and not C. This is the difference of exactly (only) A and B (which can be written, though not needed for GMAT, as ) from A and B (which can be written as )

Now how this formula is derived?

Again: when we add three groups A, B, and C some sections are counting more than once. For instance: sections 1, 2, and 3 are counted twice and section 4 thrice. Hence we need to subtract sections 1, 2, and 3 ONCE (to get one) and subtract section 4 TWICE (again to count section 4 only once).

When we subtract {Sum of Exactly 2 groups members} from A+B+C we are subtracting sum of sections 1, 2 and 3 once (so that's good) and next we need to subtracr ONLY section 4 () twice. That's it.

Now, how this concept can be represented in GMAT problem?

Example #1:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections 3 an 4 on the diagram (assuming Marketing=A, Sales=B and Vision=C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections 2 and 4);
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.

Question: Total=?

Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members.

Total=M+S+V-(MnS+SnV+SnV)+MnSnV+Neither=20+30+40-(5+6+9)+4+0=74.

Answer: 74.


Why didn't you dedcut - (2*4). I thought answer would be - 62 (20+30+40-(5+6+9)-2(4)+0 = 62. Please explain.