A specific sum over all subsets of a set

Given two positive integers $N$ and $M$ with $0<N\le M$, let $I(N,M)$ be the set which contains all natural numbers between $N$ and $M$(inclusive). For example, $I(3,6)=\{3,4,5,6\}$

We define the product of a set as the product of all elements in that set, written as P({...}) where {...} is a specific set. For example, $P(I(3,6))=3\times 4\times 5\times 6=360$

Finally, We define a function $C(N,M,K)$ as the sum of product of sets over all $I(N,M)$'s subsets which contain exactly K elements.

In other words, $C(N,M,K)={\displaystyle \sum _{S\in I(N,M),len(S)=K}P(S)}$

You are given $C(10,20,3)=549450$ and $C(100,200,10)mod({10}^{18}+3)=491824643147614094$

Find ${\displaystyle \sum _{i=1}^{500}C({10}^{20},{11}^{20},i)}$

Give your answer modulo $({10}^{18}+3)$

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[My timing: 4 s(Python)]