Question

Number of non-empty subsets of $\{1,2,3, \ldots, 12\}$ having the property that sum of the largest and smallest element is $13$ is ____.

Answer 1

According to the question, each set must contain minimum two elements, such that sum of smallest and largest element is $13$
If set contains smallest number $1$ and largest number $-12$, then we can select other elements of subset from $\{2, 3, \ldots, 12\}$
So, number of subsets are $2^{10}$
If set contains smallest number $2$ and largest number $11$ then we can select other elements of subset from $\{3 4, \ldots, 10\}$
So, number of subsets are $2^{8}$.
Similarly, we have subsets $2^{6}, 2^{4}, 2^{2}, 2^{0}$
So, total number of subsets $1+2^{2}+2^{4}+\cdots+2^{10}=1365$