Determinants & Inverses · Superchallenge problem

Let $A$ be the $5\times5$ matrix with entries $a_{ij} = \dbinom{i+j}{i}$ for $0 \le i, j \le 4$ (so its first row is $1,1,1,1,1$ and its second row is $1,2,3,4,5$). Compute $\det A$.