What is the value of determinant `|(1!,2!,3!),(2!,3!,4!),(3!,4!,5!)|` ?

Options

A. 0
B. 12
C. 24
D. 36

Answer

To find the value of the determinant \( |(1!, 2!, 3!), (2!, 3!, 4!), (3!, 4!, 5!)| \), we will follow these steps:

### Step 1: Write the Determinant
We start by writing the determinant in a more manageable form:
\[ D = \begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix} \]

### Step 2: Substitute Factorial Values
Next, we substitute the factorial values:
\[ D = \begin{vmatrix} 1 & 2 & 6 \\ 2 & 6 & 24 \\ 6 & 24 & 120 \end{vmatrix} \]

### Step 3: Calculate the Determinant
Now, we will calculate the determinant using the formula for a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a = 1, b = 2, c = 6 \), and the matrix elements are:
\[ \begin{vmatrix} e & f \\ h & i \end{vmatrix} = \begin{vmatrix} 6 & 24 \\ 24 & 120 \end{vmatrix} \]

Calculating \( ei - fh \):
\[ ei = 6 \times 120 = 720 \]
\[ fh = 24 \times 24 = 576 \]
Thus,
\[ ei - fh = 720 - 576 = 144 \]

Now, we calculate \( di - fg \):
\[ di = 2 \times 120 = 240 \]
\[ fg = 6 \times 24 = 144 \]
Thus,
\[ di - fg = 240 - 144 = 96 \]

Next, we calculate \( dh - eg \):
\[ dh = 2 \times 24 = 48 \]
\[ eg = 6 \times 6 = 36 \]
Thus,
\[ dh - eg = 48 - 36 = 12 \]

### Step 4: Substitute Back into the Determinant Formula
Now substituting back into the determinant formula:
\[ D = 1 \cdot 144 - 2 \cdot 96 + 6 \cdot 12 \]
Calculating each term:
\[ D = 144 - 192 + 72 \]
Combining these:
\[ D = 144 + 72 - 192 = 24 \]

### Final Answer
Thus, the value of the determinant is:
\[ \boxed{24} \]