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D
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If the data set S = {2, 3, 4, ..., 17}, how many subsets of S have a sum of 142?
A. 2
B. 3
C. 4
D. 5
E. 6
If the data set S = {2, 3, 4, ..., 17}, how many subsets of S have a sum of 142?
A. 2
B. 3
C. 4
D. 5
E. 6
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16 Dec 2023, 11:21
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GMAT Club Official Explanation:
If the data set S = {2, 3, 4, ..., 17}, how many subsets of S have a sum of 142?
A. 2
B. 3
C. 4
D. 5
E. 6
Firstly, let's calculate the sum of the original set: 2 + 3 + 4 + ... + 17 = (2 + 17)/2 * 16 = 152.
We need to find the number of subsets with a sum of 142, which is 10 less than the sum of S. Essentially, we need to count the subsets that total 10, because excluding them from S results in a subset summing to 142.
- The one-element subset summing to 10 is {10};
- Two-element subsets summing to 10 are {2, 8}, {3, 7}, and {4, 6}.
- The only three-element subset summing to 10 is {2, 3, 5}.
We cannot have a four-element subset summing to 10 because the smallest sum of any four elements in S is 2 + 3 + 4 + 5 = 14.
Therefore, there are five subsets summing to 142, those remaining if we remove {10}, {2, 8}, {3, 7}, {4, 6}, and {2, 3, 5}, from {2, 3, 4, ..., 17}.
Answer: D.
General Discussion
15 Dec 2023, 07:48
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Asked: If the data set S = {2, 3, 4, ..., 17}, how many subsets of S have a sum of 142?
2+3+4+... + 17 = 17*18/2 - 1 = 153 - 1 = 152
Subset S has sum 142, therefore subsets of 10 have to be removed from data set S = {2,3,4,..., 17}
Subsets to be removed = {(2,3,5),(2,8),(3,7),(4,6),(10)} : 5 subsets
5 subsets of S have a sum of 142
IMO D
2+3+4+... + 17 = 17*18/2 - 1 = 153 - 1 = 152
Subset S has sum 142, therefore subsets of 10 have to be removed from data set S = {2,3,4,..., 17}
Subsets to be removed = {(2,3,5),(2,8),(3,7),(4,6),(10)} : 5 subsets
5 subsets of S have a sum of 142
IMO D
