The value of […] is:

The correct answer is Option (3) → \(2 \times 7! \times 8! \times 9!\)

Since \(n! = n \times (n - 1)!\), each entry is a multiple of the first entry of its row:

    \(8! = 8 \times 7!,\quad 9! = 72 \times 7!\)
    \(9! = 9 \times 8!,\quad 10! = 90 \times 8!\)
    \(10! = 10 \times 9!,\quad 11! = 110 \times 9!\)

Taking out the common factors 7!, 8! and 9! from the three rows:

    \( \displaystyle \Delta = 7!\,8!\,9!\begin{vmatrix}1 & 8 & 72\\1 & 9 & 90\\1 & 10 & 110\end{vmatrix} \)

Applying \(R_2 \to R_2 - R_1\) and \(R_3 \to R_3 - R_1\):

    \( \displaystyle \Delta = 7!\,8!\,9!\begin{vmatrix}1 & 8 & 72\\0 & 1 & 18\\0 & 2 & 38\end{vmatrix} \)

Expanding along the first column:

    \(\Delta = 7!\,8!\,9!\,(1 \times 38 - 18 \times 2)\)
    \(\Delta = 2 \times 7! \times 8! \times 9!\)