Does every integer occur finitely many times and in what positions in Pascal's triangle? Given number n, does it occurs finitely many times in Pascal's triangle? In what positions?
 A: $n$ only appears within the first $n+1$ rows.  After that, the numbers are all bigger than $n$.
I think the most common number known to appear is 3003 which appears eight times.
A: How to reverse the $n$ choose $k$ formula? Probably gives powerful answers to this question and more. It says that n = X must occur within a bounded region of rows (and diagonals I think) in Pascal's triangle for fixed k, and that you can brute force where it should be. If you are using a computer to count, just iterate through the k's and use the information there, it should somehow reduce the search depth.
Note that every number does occur somewhere in the triangle, because the second diagonal just counts up. So while you may not be able to count it easily, every number appears once. And thanks to Michael's answer, it is clear that a given number has a bound in terms of the rows. The diagonals ascend, so then any number can only appear as much as its value with respect to the symmetry of the table... not much of a bound, but better than just it appears somewhere in the triangle a bunch of times above the rows.
