HMMT November 2011 H5

For any finite sequence of positive integers $\pi$, let $S(\pi)$ be the number of strictly increasing subsequences in $\pi$ with length 2 or more. For example, in the sequence $\pi=\{3,1,2,4\}$, there are five increasing sub-sequences: $\{3,4\},\{1,2\},\{1,4\},\{2,4\}$, and $\{1,2,4\}$, so $S(\pi)=5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order $\pi$ from left to right in her hand. Determine

$$\sum_{\pi} S(\pi)$$

where the sum is taken over all possible orders $\pi$ of the card values.