Explanation

(b)
Given determinant is
$\mathrm{\Delta} = \left| \begin{matrix}
15! & 16! & 17! \\
16! & 17! & 18! \\
17! & 18! & 19!
\end{matrix} \right|$
Applying $C_{2} \rightarrow C_{2} - C_{1},\ C_{3} \rightarrow C_{3} - C_{2}$
$\mathrm{\Delta} = \left| \begin{matrix}
15! & 15 \times 15! & 16 \times 16! \\
16! & 16 \times 16! & 17 \times 17! \\
17! & 17 \times 17! & 18 \times 18!
\end{matrix} \right|$
= (15!)(16!)(17!)$\left| \begin{matrix}
1 & 15 & 16 \times 16 \\
1 & 16 & 17 \times 17 \\
1 & 17 & 18 \times 18
\end{matrix} \right|$
Applying $R_{1} \rightarrow R_{1} - R_{2},R_{2} \rightarrow R_{2} - R_{3}$
= (15!)(16!)(17!)$\left| \begin{matrix}
0 & - 1 & - 33 \\
0 & - 1 & - 35 \\
1 & 17 & 18 \times 18
\end{matrix} \right|$
=$2 \times (15!)(16!)(17!)$