Theres a problem from the The ICPC 2018 — Vietnam Central Provincial Contest that supposedly involves some combinatorics (The problem is here)

I originally planned to count the ways to break N/K into sum of at most M numbers, and then count the number of permutations of the children for each ways, and then use binpow to calculate total_ways % 1e9+7 (probably the first thing you would think of when you see the problem). But the limit was so high that i will have to choose between ridiculously long running time or using bignum (which can cause either TLE too (or MRE), whatever comes first.)

So it would be better if i could have an O(1) or an O(log N/K) solution for this problem.