Compute `(4!)(2!)`. Is `(4!)(2!)=8!` ?

Answer

To solve the problem, we need to compute \( (4!)(2!) \) and determine if it is equal to \( 8! \).

### Step-by-Step Solution:

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

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

3. **Multiply \( 4! \) and \( 2! \)**:
\[ (4!)(2!) = 24 \times 2 = 48 \]

4. **Now, calculate \( 8! \)**:
\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

5. **Break down \( 8! \)**:
\[ 8! = 8 \times 7 \times 6 \times 5 \times (4 \times 3 \times 2 \times 1) = 8 \times 7 \times 6 \times 5 \times 4! \]

6. **Express \( 8 \) in terms of \( 2 \)**:
\[ 8 = 2 \times 2 \times 2 \]
So we can rewrite \( 8! \) as:
\[ 8! = (2 \times 2 \times 2) \times 7 \times 6 \times 5 \times 4! \]

7. **Factor out \( 2! \)**:
\[ 8! = 7 \times 6 \times 5 \times (4!) \times (2 \times 2) = 7 \times 6 \times 5 \times (4!) \times (2!) \]

8. **Conclusion**:
Since \( (4!)(2!) = 48 \) and \( 8! \) is a much larger number, we conclude that:
\[ (4!)(2!) \neq 8! \]

### Final Answer:
\[ (4!)(2!) = 48 \quad \text{and} \quad (4!)(2!) \neq 8! \]

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