question_answer

For a fixed positive integer n, if \[D=\left| \begin{matrix}    n! & (n+1)! & (n+2)!  \\    (n+1)! & (n+2)! & (n+3)!  \\    (n+2)! & (n+3)! & (n+4)!  \\ \end{matrix} \right|,\] then \[\left[ \frac{D}{{{(n!)}^{3}}}-4 \right]\] is divisible by ________.

A)  \[({{n}^{2}}+4n+2)\]            

B)  \[\left( 3{{n}^{2}}+2n+1 \right)\]

C)  \[\left( {{n}^{2}}+4n+5 \right)\]  

D)  \[\left( {{n}^{2}}+4n+9 \right)\]

E)  None of these