There's an array (let's call it $$$d$$$) used in the standard algorithm for finding the longest increasing subsequence. We are interested in its first five elements. Recall that the input is a permutation. We see that $$$\mathit{choose} (40, 5) = 658\,008$$$. So perhaps we can devise a dynamic programming solution. The state can be the first five numbers of the array $$$d$$$.