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A = {1, 2, 3, …., 12}. A1, A2, A3, …., An are all the s

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A = {1, 2, 3, …., 12}. A1, A2, A3, …., An are all the s

by Max@Math Revolution » Tue Jan 07, 2020 11:37 pm
[GMAT math practice question]

A = {1, 2, 3, ...., 12}. A1, A2, A3, ...., An are all the subsets of A with m elements. If ak is the summation of Ak, what is a1 + a2 +....+ an?

1) m = 3
2) n = 220
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Source: — Data Sufficiency |
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by Max@Math Revolution » Fri Jan 10, 2020 12:54 am
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Condition 1)

A1, A2, ..., An are subsets of set A = {1, 2, ..., 12} with three elements.
Then the number of subsets Ai's containing 1 is 11C2 = 11*10/1*2 = 55, which is the number of cases to choose 2 elements out of 11 elements.
Thus, when we calculate a1 + a2 + ... + ak, each element is added 55 times.
Then we have a1 + a2 + ... + ak = 55(1 + 2 +...+ 12) = 4290.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
12C1 = 12/1 = 12, 12C2 = 12*11/1*2 = 66 and 12C3 = 12*11*10/1*2*3 = 220.
Then 12Cm = 12C3 and m = 3.
Thus, condition 2) is equivalent to condition 1), and it is sufficient.

Therefore, D is the answer.
Answer: D


Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).
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