Explanation

(c)
Let $\mathrm{\Delta} = \left| \begin{matrix}
10! & 11! & 12! \\
11! & 12! & 13! \\
12! & 13! & 14!
\end{matrix} \right|$
$= 10!11!12!\left| \begin{matrix}
1 & 11 & 11 \times 12 \\
1 & 12 & 12 \times 13 \\
1 & 13 & 13 \times 14
\end{matrix} \right|$
Applying $R_{2} \rightarrow R_{2} - R_{1}\ and\ R_{3} \rightarrow R_{3} - R_{1}$
$= 10!11!12!\left| \begin{matrix}
1 & 11 & 11 \times 12 \\
0 & 1 & 24 \\
0 & 2 & 50
\end{matrix} \right|$
$= (10!11!12!)(50 - 48)$
$= 2 \bullet (10!11!12!)$