My homework is to calculate the determinant of the matrix: $$ \det(A) = \begin{vmatrix} \binom{n}{0} & \binom{n}{1} & \cdots & \binom{n}{r} \\\ \binom{n+1}{0} & \binom{n+1}{1} & \cdots & \binom{n+1}{r} \\\ \vdots & \vdots & \ddots & \vdots \\\ \binom{n+r}{0} & \binom{n+r}{1} & \cdots & \binom{n+r}{r} \end{vmatrix} = D(r,n) $$ and following is my proof:
Using the identity $\binom{m}{k} + \binom{m}{k+1} = \binom{k+1}{m+1}$, then after transforming A by a sequence of operators on rows $$ L_{r+1} \leftarrow L_{r+1} - L_{r}, \: L_{r} \leftarrow L_{r} - L_{r-1}, \dots, \: L_2 \leftarrow L_2 - L_1 $$ (the last row is affected first, then the second row...)
we have $$ \begin{vmatrix} \binom{n}{0} & \binom{n}{1} & \cdots & \binom{n}{r} \\\ \binom{n+1}{0} & \binom{n+1}{1} & \cdots & \binom{n+1}{r} \\\ \vdots & \vdots & \ddots & \vdots \\\ \binom{n+r}{0} & \binom{n+r}{1} & \cdots & \binom{n+r}{r} \end{vmatrix} = \begin{vmatrix} \binom{n}{0} & \binom{n}{1} & \cdots & \binom{n}{r} \\\ 0 & \binom{n}{0} & \cdots & \binom{n}{r-1} \\\ \vdots & \vdots & \ddots & \vdots \\\ 0 & \binom{n+r-1}{0} & \cdots & \binom{n+r-1}{r-1} \end{vmatrix} = \binom{n}{0} \begin{vmatrix} \binom{n}{0} & \cdots & \binom{n}{r-1} \\\ \vdots & \ddots & \vdots \\\ \binom{n+r-1}{0} & \cdots & \binom{n+r-1}{r-1} \end{vmatrix} $$ Or $$ D(r,n) = D(r-1,n-1) $$ By recurrence $$ D(r,n) = D(0,n-r) = \binom{n-r}{0} = 1 $$ The grader gave me simply "Wrong" as the feedback but I still cannot find where the error is.
Thanks for any help.