Evaluate: `=|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !|` .

Explanation

To evaluate the determinant

\[ D = \begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 12! \\ 12! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} \]

we can follow these steps:

### Step 1: Rewrite the Factorials
We can express the factorials in terms of \(10!\):

- \(11! = 11 \times 10!\)
- \(12! = 12 \times 11! = 12 \times 11 \times 10!\)
- \(12! = 12 \times 11!\)
- \(13! = 13 \times 12!\)
- \(14! = 14 \times 13!\)

Thus, we can rewrite the determinant as:

\[ D = \begin{vmatrix} 10! & 11 \times 10! & 12 \times 11 \times 10! \\ 11 \times 10! & 12 \times 11 \times 10! & 12 \times 11 \times 10! \\ 12 \times 11 \times 10! & 12 \times 11 \times 10! & 13 \times 12 \times 11 \times 10! \\ 12 \times 11 \times 10! & 13 \times 12 \times 11 \times 10! & 14 \times 13 \times 12 \times 11 \times 10! \end{vmatrix} \]

### Step 2: Factor Out \(10!\)
Now, we can factor \(10!\) out of each column:

\[ D = 10!^3 \begin{vmatrix} 1 & 11 & 12 \times 11 \\ 11 & 12 \times 11 & 12 \times 11 \\ 12 \times 11 & 12 \times 11 & 13 \times 12 \\ 12 \times 11 & 13 \times 12 & 14 \times 13 \end{vmatrix} \]

### Step 3: Simplify the Matrix
Now, we can simplify the determinant. Notice that the second and third columns have similar terms. We can perform row operations to simplify it further.

1. Subtract the first row from the second and third rows:

\[ D = 10!^3 \begin{vmatrix} 1 & 11 & 12 \times 11 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2 \end{vmatrix} \]

### Step 4: Calculate the Determinant
Now, we can calculate the determinant of the simplified matrix:

\[ D = 10!^3 \cdot 1 \cdot \begin{vmatrix} 1 & 12 \times 11 \\ 1 & 2 \end{vmatrix} \]

Calculating this 2x2 determinant:

\[ = 10!^3 \cdot (1 \cdot 2 - 1 \cdot 12 \times 11) = 10!^3 \cdot (2 - 12 \times 11) \]

### Step 5: Final Calculation
Now, substituting the values:

\[ = 10!^3 \cdot (2 - 132) = 10!^3 \cdot (-130) \]

### Final Answer
Thus, the value of the determinant is:

\[ D = -130 \cdot 10!^3 \]