$N(n)=n \prod_{r=1}^{2023}\left(n^2-r^2\right)=n \cdot\left[\prod_{r=1}^{2023}(n-r)\right]\left[\prod_{r=1}^{2023}(n+r)\right]$
but when $\mathrm{n}=2024$ ( given)
$$
\begin{aligned}
& \Rightarrow \quad N(2024)=2024[(2023)(2022)(2021) \ldots 3.2 .1] \\
& {[(2025)(2026) \ldots(4046)(4047)]} \\
&
\end{aligned}
$$

Now rearranging above we get,
$$
\begin{aligned}
& \Rightarrow \quad N(2024)=(4047)(4026) \ldots(2025)(2024)(2023) \ldots 3.2 .1 \\
& =(4047) !
\end{aligned}
$$
but we have to find ${ }^N C_{N-1}$ when $n=2024$
Now ${ }^N C_{N-1}=\frac{N !}{(N-1) !(N-(N-1)) !}=\frac{N(N-1) !}{(N-1) ! 1 !}$
$$
\begin{aligned}
& { }^N C_{N-1}=N \\
& { }^N C_{N-1}=(4047) !\{\text { when } n=2024
\end{aligned}
$$