Question 13
Let
$$N=\sum_{k_1=0}^{2025}\sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\cdots\sum_{k_{2098}=0}^{k_{2097}}
\frac{n!}{(n-k_1)!(k_1-k_2)!\cdots(k_{2097}-k_{2098})!\,k_{2098}!}.$$
Find the remainder when $N$ is divided by $3000$.
The supplied scan includes the handwritten annotations $n=2025$ and the terminal factor $k_{2098}!$; both are retained in the transcription above and shown in the original excerpt below.
$$N=\sum_{k_1=0}^{2025}\sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\cdots\sum_{k_{2098}=0}^{k_{2097}}
\frac{n!}{(n-k_1)!(k_1-k_2)!\cdots(k_{2097}-k_{2098})!\,k_{2098}!}.$$
Find the remainder when $N$ is divided by $3000$.
The supplied scan includes the handwritten annotations $n=2025$ and the terminal factor $k_{2098}!$; both are retained in the transcription above and shown in the original excerpt below.