Question

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

Answer 1

Answer: 1365

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 $\{2,3,\dots ,12\}$
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 $\{34,\dots ,10\}$
So, number of subsets are $2^8$.
Similarly, we have subsets $2^6,2^4,2^2,2^0$
So, total number of subsets $1+2^2+2^4+\cdots +2^{10}=1365$