Here's an interesting combinatoric problem I have no clue about:
How many permutations of $1, 2, 3, ..., 19, 20$ exist such that the $5$th element is $7$, and the $10$th element is $14$, and there exists at least one pair of $x, y$ such that $x < y$, $A_x > y$ and $A_y > x$ where $A$ is the permutation.
I have been thinking of this for a while without many ideas. I think if we can somehow count all the permutations, we should be able to remove those that don't satisfy the other constraint.
Thanks!