The value of the determinant $Δ=\begin{vmatrix}1!& 2!& 3!\\2!&3!& 4!\\3!& 4! & 5!\end{vmatrix}$ is :

$4!$

The correct answer is option (3) → $4!$

$Δ=\begin{vmatrix}1!& 2!& 3!\\2!&3!& 4!\\3!& 4! & 5!\end{vmatrix}=1!2!3!\begin{vmatrix}1&2&2×3\\1&3& 3×4\\1&4&4×5\end{vmatrix}$

$=12\begin{vmatrix}1&2&6\\1&3& 12\\1&4&20\end{vmatrix}$

$R_3→R_3-R_2,R_2→R_2-R_1$

$=12\begin{vmatrix}1&2&6\\0&1& 6\\0&1&8\end{vmatrix}=24=4!$