Find the Value of the Determinant

If $M(n)$ is the $n \times n$ matrix $M(n) \; = \; \left( \begin{array}{cccc} \binom{2000}{0} & \binom{2000}{1} & \cdots & \binom{2000}{n-1} \\ \binom{2001}{0} & \binom{2001}{1}& \cdots & \binom{2001}{n-1} \\ \vdots & \vdots & \ddots & \vdots \\ \binom{2000+n-1}{0} & \binom{2000+n-1}{1} & \cdots & \binom{2000+n-1}{n-1} \end{array} \right)$ then, since $\binom{n+1}{k+1} - \binom{n}{k+1} = \binom{n}{k}$ we see that, subtracting the $(n-1)$ th row from the $n$ th, then the $(n-2)$ nd from the $(n-1)$ th, the $(n-3)$ rd from the $n-2)$ nd, and so on up, finally subtracting the $1$ st row from the $2$ nd, that $\mathrm{det}\,M(n) \; = \; \mathrm{det}\,\left( \begin{array}{cccc} 1 & \binom{2000}{1} & \cdots & \binom{2000}{n-1} \\ 0 & & & \\ \vdots & & M(n-1) & \\ 0 & & & \end{array}\right) \; =\; \mathrm{det}\,M(n-1)$ and hence $\mathrm{det}\,M(19) = \mathrm{det}\,M(1) = \boxed{1}$ .