What is ncr?

I don't know if this is fast enough for your case but it might be:

1. Write a brute force program to compute all factorials of the form $$$y!$$$ such that $$$y$$$ is a multiple of $$$X$$$ and put them in a static array $$$Y$$$ in your real program.
2. To compute any factorial of the form $$$z!$$$, find the largest element $$$e$$$ of the form $$$k!$$$ in $$$Y$$$ such that $$$k \leq z$$$, and then iterate from $$$k+1$$$ to $$$z$$$ and multiply all these numbers by $$$e$$$.

The complexity of calculating any NCr would be $$$O(X)$$$ and the memory complexity (for the static array) would be $$$O(\frac{N}{X})$$$, so maybe $$$X = 10^6$$$ is enough.

I don't understand why this problem is so hard. I think it is fairly easy. Notice the constraint $$$n-r \le 10^6$$$. You can easily calculate value of $$$C(n,r)$$$ in $$$O(r)$$$. For further explanation, see hint.

Lastly, you need to remember that $$$C(n,r) = C(n,n-r)$$$, hence if $$$n-r \le 10^6$$$, then you should calculate $$$C(n,n-r)$$$ instead of calculating $$$C(n,r)$$$.

You can refer to editorial of this problem.