The value of the determinant $\begin{vmatrix}10! & 11! & 12!\\11! & 12! & 13!\\12! & 13! & 14!\end{vmatrix},$ is

$2(10!11!12!)$

The correct answer is option (3) : $2(10!11!12!)$

We have,

$\begin{vmatrix}10! & 11! & 12!\\11! & 12! & 13!\\12! & 13! & 14!\end{vmatrix}$

$=10!×11!×12!\begin{vmatrix}1 & 11 & 132\\1 & 12 & 156\\1 & 13 & 182\end{vmatrix}$

$=10!×11!×12!\begin{vmatrix}1 & 11 & 132\\0 & 1 & 24\\0 & 1 & 26\end{vmatrix}$    $\begin{bmatrix}Applying \, R_2→R_2-R_1,\\R_3→R_3-R_2\end{bmatrix}$

$=2(10!×11!×12!)$