Derangements of $6$ elements

Recurrence. $D_n=(n-1)(D_{n-1}+D_{n-2})$. $D_4=9$, $D_5=44$, so $D_6=5(44+9)=5\cdot53=265$.

Check via formula. $D_6=720(1-1+1/2-1/6+1/24-1/120+1/720)=720(1/2-1/6+1/24-1/120+1/720)$.

$\frac12-\frac16+\frac1{24}-\frac1{120}+\frac1{720}=\frac{360-120+30-6+1}{720}=\frac{265}{720}$.

So $D_6=720\cdot265/720=265$. ✓

Conclusion. $D_6=265$.