Let `alpha=((4!)!)/((4!)^(3!))` and `beta=((5!)!)/((5!)^(4!))` .Then:

Options

A. `alpha not N" and "beta not N`
B. `alpha not N" and "beta in N`
C. `alpha in N" and "beta!in N`
D. `alpha in N" and "beta in N`

Answer

To solve the problem, we need to compute the values of \( \alpha \) and \( \beta \) and analyze their properties.

### Step-by-Step Solution:

1. **Calculate \( 4! \)**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

2. **Calculate \( 3! \)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]

3. **Calculate \( \alpha \)**:
\[ \alpha = \frac{(4!)!}{(4!)^{3!}} = \frac{24!}{24^{6}} \]

4. **Calculate \( 5! \)**:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

5. **Calculate \( 4! \)** again (already calculated):
\[ 4! = 24 \]

6. **Calculate \( \beta \)**:
\[ \beta = \frac{(5!)!}{(5!)^{4!}} = \frac{120!}{120^{24}} \]

7. **Combine \( \alpha \) and \( \beta \)**:
\[ \alpha \beta = \left(\frac{24!}{24^{6}}\right) \left(\frac{120!}{120^{24}}\right) \]

8. **Determine if \( \alpha \) and \( \beta \) are natural numbers**:
- \( \alpha \) is a natural number because \( 24! \) is divisible by \( 24^6 \).
- \( \beta \) is also a natural number because \( 120! \) is divisible by \( 120^{24} \).

### Final Result:
Both \( \alpha \) and \( \beta \) are natural numbers.