We need to calculate how many times each number from the input appears as the largest of $K$ numbers in every combination. If we sort the numbers in ascending order, we can see that this number is going to be maximal in an array of $K$ numbers if all other numbers are located to its left. Therefore, the $i^{\text{th}}$ number is maximal the exact number of times as the number of ways to choose $K-1$ numbers out of the first $i$ numbers. If we denote the number of ways to choose $K$ numbers out of $N$ numbers as $f(n,k)$, then it is easy to see that $f(n,k)=f(n-1,k)+f(n-1,k-1)$.

Using this relation, we can calculate each $f(n,k)$.

The solution is therefore the sum of the product of the corresponding $f(i,k-1)$ and $v[i]$, where $v[i]$ is the value on that location.