Three couples are sitting in a row. Compute the number of arrangements in which no person is sitting next to his or her partner. Answer is 240.
From wikipedia, This problem is called the menage problem have a fourmula called Touchard's formula:
Let $M_n$ denote the number of seating arrangements for n couples. Touchard (1934) derived the formula
$M_n = 2 \cdot n! \sum_{k=0}^n (-1)^k \frac{2n}{2n-k} {2n-k\choose k} (n-k)!$
How does one mnage to prove it