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A Messy Dinner

 Published on Sunday, 7th February 2021, 07:00 am; Solved by 181

Problem 746

$n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large circular table with $4n$ seats such that men and women alternate.

Let $M(n)$ be the number of ways the families can be seated such that none of the families were seated together. A family is considered to be seated together only when all the members of a family sit next to each other.

For example, $M(1)=0$, $M(2)=896$, $M(3)=890880$ and $M(10) \equiv 170717180 \pmod {1\,000\,000\,007}$

Let $S(n)=\displaystyle \sum_{k=2}^nM(k)$

For example, $S(10) \equiv 399291975 \pmod {1\,000\,000\,007}$

Find $S(2021)$. Give your answer modulo $1\,000\,000\,007$.