Find the number of subsets of {1, 2, 3, ..., 10}, where the sum of the smallest element and the largest element is equal to 11.

Guest Jun 21, 2021

#3**+2 **

Let's elaborate. We define an $n$--$x$ set to be a set having $n$ as the minimum element, and $x$ as the maximum element such that $n+x=11$. E.g., $\{4, 5, 7\}$ is a 4--7 set. There are five possibilities according to the extreme elements.

(i) 5--6 sets: There's only one set $\{5, 6\}$.

(ii) 4--7 sets: Besides 4 and 7 that need to belong, the numbers 5 and 6 can be chosen to be included or excluded independently. So there are $2^2$ sets of this kind.

(iii) 3--8 sets: Besides 3 and 8 that need to belong, the numbers 4, 5, 6, and 7 can be chosen to be included or excluded independently. So there are $2^4$ sets of this kind.

(iv) 2--9 sets: Besides 2 and 9 that need to belong, the numbers 3, 4, 5, 6, 7, and 8 can be chosen to be included or excluded independently. So there are $2^6$ sets of this kind.

(v) 1--10 sets: Besides 1 and 10 that need to belong, the numbers 2, 3, 4, 5, 6, 7, 8, and 9 can be chosen to be included or excluded independently. So there are $2^8$ sets of this kind.

Adding numbers from all five possibliities give the answer.

Bginner Jun 21, 2021