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31 Dec 2023, 00:58
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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
A. 2
B. 3
C. 4
D. 5
E. 6
31 Dec 2023, 00:59
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Official Solution:
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: \(2 + 3 + 4 + ... + 17 = \frac{2 + 17}{2} * 16 = 152\).
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
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: \(2 + 3 + 4 + ... + 17 = \frac{2 + 17}{2} * 16 = 152\).
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
24 Apr 2024, 22:25
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I think this is a high-quality question.
