Bonus for p550

Given two natural numbers $a$ and $b$ with $1<a\le b$, let $J(a,b,N)$ be the set which contains all natural numbers not greater than N and not divisible by any number between $a$ and $b$(inclusive). For example, $J(4,6,10)=\{1,2,3,7,9\}$

4 can divide 4 and 8

5 can divide 5 and 10

6 can divide 6

Hence 4,5,6,8 and 10 will not appear in the set which $J(4,6,10)$ represents. Although they are all not greater than 10.

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, $P(J(4,6,10))=1\times 2\times 3\times 7\times 9=378$

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

In other words, $D(a,b,N,K)={\displaystyle \sum _{S\in J(a,b,N),len(S)=K}P(S)}$

You are given $D(5,8,20,3)=46035$ and $D(7,12,3000,10)mod({10}^{18}+3)=736056499082019203$

Find ${\displaystyle \sum _{i=1}^{150}D(11,20,{11}^{20},i)}$

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

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[My timing: 18 s(Python)]