The value of $\begin{vmatrix}7!&8!&9!\\8!&9!&10!\\9!&10!&11!\end{vmatrix}$ is:

$2 × 7! × 8! × 9!$

The correct answer is Option (3) → $2 × 7! × 8! × 9!$

$\left|\begin{matrix} 7! & 8! & 9!\\ 8! & 9! & 10!\\ 9! & 10! & 11! \end{matrix}\right|$

$= \left|\begin{matrix} 7! & 8\cdot7! & 9\cdot8\cdot7!\\ 8\cdot7! & 9\cdot8\cdot7! & 10\cdot9\cdot8\cdot7!\\ 9\cdot8\cdot7! & 10\cdot9\cdot8\cdot7! & 11\cdot10\cdot9\cdot8\cdot7! \end{matrix}\right|$

$=7!\,8!\,9! \left|\begin{matrix} 1 & 1 & 1\\ 1 & 9 & 10\\ 1 & 10 & 110 \end{matrix}\right|$

$R_2\rightarrow R_2-R_1,\; R_3\rightarrow R_3-R_1$

$=7!\,8!\,9! \left|\begin{matrix} 1 & 1 & 1\\ 0 & 8 & 9\\ 0 & 9 & 109 \end{matrix}\right|$

$=7!\,8!\,9! \left|\begin{matrix} 8 & 9\\ 9 & 109 \end{matrix}\right|$

$=7!\,8!\,9!(8\cdot109-9\cdot9)$

$=7!\,8!\,9!(872-81)$

$=7!\,8!\,9!\times2$

Final answer: $2\times7!\times8!\times9!$