I don’t know about you, but when I see formulas for sets that look like P(A) + P(B) + P(C) – 2P(A n B) – 2P(A n C) – 2P(B n C) + 3P(A n B n C), it takes me a minute for my brain to recall exactly what all these signs and symbols mean.
Even if we’re die-hard GMAT-ers, we’re just not used to seeing that many sets questions on practice tests, so while we know n = intersection and u = union, these formulas are just not easy to recall or employ.
Rather than memorize all the possible versions of this formula (to find the number of items in exactly one, two, and three sets), the “magical” formula you really need to know is ABC-2-23. That little code helps us to easily remember the formula to find the Overall Total of a set with three categories by distilling down this monster:
Overall Total = (# in Category 1) + (# in Category 2) + (# in Category 3) – (# who fit 2 categories) – 2(# who fit all three) + (# who fit none).
If we call each category A, B, and C (which most versions of the sets formula do), then it becomes much simpler:
Total = A + B + C – (2) – 2(3) + None
Let’s try ABC-2-23 out!
Of the Baskin Robbins ice cream stores in the county, 40 stock Rocky Road ice cream, 50 stock Pistachio ice cream, and 35 stock Jamocha ice cream. If each store stocks at least one of these flavors, how many Baskin Robbins stores are in the county?
(1) 42 of the stores stock exactly two of these flavors.
(2) None of the stores stock all three flavors.
To find the total number of stores, we need to know how many stores overlap ice cream flavors. Statement (1) tells us that 42 of the stores stock exactly two. Let’s lay out ABC-2-23:
Total = A + B + C – (# who fit 2 categories) – 2(# who fit all three) + (# who fit none).
The question-stem tells us that A = 40, B = 50, C = 35, and that 0 = None. So we have two pieces of missing information: the number of stores who sell 2 of the ice creams, and the number of stores who sell 3 of the ice creams (the “2” and the “3” from our mnemonic):
Total = 40 + 50 + 35 – (42) – 2(x) + 0
Total = 83 – 2x
We need to know the number of stores who stock all three flavors in order to be sufficient. Statement (2) is insufficient by itself for the same reason. It allows us to fill in our formula:
Total = 40 + 50 + 35 – (# who fit 2 categories) – 2(0) + 0
Total = 125 – x
Combined, we have two equations with the same two variables.
Total = 40 + 50 + 35 – (42) – 2(0) + 0
Total = 125 – (42) – 0 + 0
Total = 83
The correct response is (C). Sufficient.
The only slightly tricky thing about the mnemonic device ABC-2-23 is remembering that the middle “2” and the final “3” refer to the category, and not an actual integer. It’s important to remember that “A” stands for everything in the first category, including any overlaps with other categories. It does not mean “category A only.”
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing advice to help students better prepare for the GMAT and the SAT.