Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation?
$$\binom{p^2}{q}-\binom{q}{p}=1$$
This question was proposed for Romanian national olympiad in 2006. There is a very long solution in the AOPS website for the problem. I was wondering if there is a simpler solution!