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2021/22 AHSMC Part 1 Problem 16

Problem 16 of 16

The number of subsets of {15,16,17,…,57,58}\{15, 16, 17, \ldots, 57, 58\} having the sum of their elements equal to 15581558 is

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Solution

Since 15+16+⋯+58=58⋅592−14⋅152=160615 + 16 + \cdots + 58 = \dfrac{58 \cdot 59}{2} - \dfrac{14 \cdot 15}{2} = 1606 the problem is equivalent to finding the number of subsets of {15,16,…,58}\{15, 16, \ldots, 58\} whose sum of elements is 1606−1558=481606 - 1558 = 48. These subsets are the following: {48}, {15,33}, {16,32}, {17,31}, {18,30}, {19,29}, {20,28}, {21,27}, {22,26}, {23,25}, {15,16,17}. \{48\},\ \{15,33\},\ \{16,32\},\ \{17,31\},\ \{18,30\},\ \{19,29\},\ \{20,28\},\ \{21,27\},\ \{22,26\},\ \{23,25\},\ \{15,16,17\}. The answer is (E).

University of Alberta