Differences between the two formulas for advanced overlapping sets

Question

Hello everyone!

I am currently trying to understand the two formulas for the advances overlapping sets (Venn diagrams of three sets), and I don't really understand the two formulas.

Tot = A + B + C - (sum of 2 group overlaps) + (all 3) + neither 
Tot = A + B + C - (sum of EXACTLY 2 group overlaps) - 2 (all 3) + neither
What is the main difference between the two? Can both be applied to the same problem?

Thank you so much for your time and attetion!

$!vakumar.m · Answer

1.  The two formulas are used depending upon the information given in the question. 
If info on sum of 2 groups is given then use the first formula and
if info on sum of exactly 2 group is given then use the second formula

( Refer Diagram attached )
2. We have three types of overlap=> two set overlap - 3 (region 1, 2 and 3)
 exactly 2 group overlap - 3 (region x,y and z) 
 three set overlap - 1 (all three)

3.  Formula 1 - info on sum of 2 group is given 
Tot = A + B + C - (sum of 2 group overlaps) + (all 3) + neither 
Note: 
In the Sum of A + B + C, the sum of two group overlap region i.e sum of 1, 2 and 3 is counted twice 
and All three region is counted thrice (once with each set A, B and C)
so, we subtract sum of two group overlap region (1+2+3) once 
but when we perform A+B+C - sum of 2 group overlaps then all three region is not counted even once (note that all three region is counted thrice in sum of A+B+C and thrice in sum of 2 group overlap, hence get subtracted), so we need to add all three region once in the formula above.

4.  Formula 2 - info on sum of exactly 2 group is given 
Tot = A + B + C - (sum of EXACTLY 2 group overlaps) - 2 * (all 3) + neither
Here, in the Sum of A + B + C, the sum of exactly 2 group overlap region i.e x, y and z is counted twice 
and All three region is counted thrice
so, we subtract sum of two group overlaps (x+y+z) once and all three region twice

GmatKnightTutor · Answer

You may want to check the gmatclub in house math guide. These two formulas are mentioned in it.

ReedArnoldMPREP · Answer

You determine which of these to use by determining *how the problem gives you information about the 'two group' overlap.* If the problem says "10 things were A and B," that also includes some that were A, B, and C. If the problem says "10 of the things are ONLY A and B," that does not include the C.

When building the three overlapping set diagram, the thing to really keep track of is 'how many times each section gets counted,' and make your math reflect that. Memorizing these formulas is fine, but you don't really even need to if you just get good at building your model well.

This blog post might help:  https://www.manhattanprep.com/gmat/blog/how-to-handle-3-group-overlapping-sets-on-the-gmat/

ScottTargetTestPrep · Answer

In the first formula, to calculate "sum of 2 group overlaps", you should add the number of elements that belong to A and B, A and C, and B and C; whereas in the second formula, to calculate "sum of exactly 2 group overlaps", you should add A and B but not C, A and C but not B, and B and C but not A.

Most questions can be solved using either formula, but usually, depending on the information given in the question, it makes sense to choose one of the formulas over the other. For instance, if the question gives you the number of elements in "exactly 2 group overlaps" and if you know the number of elements in the triple overlap, you can determine the number of elements in "2 group overlaps" by adding three times the triple overlap to the exactly 2 groups overlaps, but clearly it makes more sense to use the formula which mentions "exactly 2 group overlaps".

Differences between the two formulas for advanced overlapping sets

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