Question: A={1,2,3,4,5} is given. If there are 1,2,…,n subsets of A, let A1, A2 , A3 ,… , An be the sums of the elements of the corresponding subsets of A. What is A1+ A2,+ A3+… + An?

1. 200
2. 210
3. 230
4. 240
5. 250

Correct Answer: D

Solution and Explanation:
Approach Solution 1:

A has 2^5= 32 subsets, so n=32. Each element of A is contained in 2^5−1=24= 16 subsets of A (We fix one element and find the number of subsets of the remaining elements). So, A1 + A2 + A3 + … + A32 = 16∗(1+2+3+4+5)= 240.

Approach Solution 2:

The subsets of A={1,2,3,4,5} are as follows.

S1={},
S2={1},
S3={2},
S4={3},
S5={4},
S6={5},
S7={1,2},
S8={1,3},
S9={1,4},
S10={1,5},
S11={2,3},
S12={2,4},
S13={2,5},
S14={3,4},
S15={3,5},
S16={4,5},
S17={1,2,3},
S18={1,2,4},
S19={1,2,5},
S20={1,3,4},
S21={1,3,5},
S22={1,4,5},
S23={2,3,4},
S24={2,3,5},
S25={2,4,5},
S26={3,4,5},
S27={1,2,3,4},
S28={1,2,3,5},
S29={1,2,4,5},
S30={1,3,4,5},
S31={2,3,4,5},
S32={1,2,3,4,5}.

The half of 32 sets, 16 sets have an element 1, 2, 3, 4 and 5 respectively.
When we calculate A1+A2+⋯+A32, 1 happens 16 times, 2 happens 16 times, ... and 5 happens 16 times.
That's why we have A1+A2+⋯+A32=16∗1+16∗2+...+16∗5=16(1+2+3+4+5)= 16∗15= 240.

“A={1,2,3,4,5} is given. If there are 1,2,…,n subsets of A, let A1”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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