Let
`N=|(10!,11!,12!),(11!,12!,13!),(12!,13!,14!)|`
Then `N/((10!)(11!)(12!))=`______

Answer

To solve the problem, we need to calculate the determinant \( N = \begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} \) and then find \( \frac{N}{(10!)(11!)(12!)} \).

### Step 1: Factor out the common factorials
We notice that \( 10! \), \( 11! \), and \( 12! \) are common in each row. We can factor them out:

\[ N = 10! \cdot 11! \cdot 12! \cdot \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix} \]