The value of `8! div 4!` is :

Options

A. 4
B. 16
C. 2
D. 1680

Answer

To solve the problem of finding the value of \( \frac{8!}{4!} \), we can follow these steps:

### Step 1: Understand Factorials
Factorials are defined as the product of all positive integers up to a given number. For example:
- \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \)

### Step 2: Write the Factorials
We need to express \( 8! \) and \( 4! \):
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 4! = 4 \times 3 \times 2 \times 1 \)

### Step 3: Substitute the Factorials into the Expression
Now we substitute these into our expression:
\[ \frac{8!}{4!} = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4!} \]

### Step 4: Cancel the \( 4! \)
Since \( 4! \) appears in both the numerator and the denominator, we can cancel it out:
\[ \frac{8 \times 7 \times 6 \times 5 \times 4!}{4!} = 8 \times 7 \times 6 \times 5 \]

### Step 5: Calculate the Remaining Product
Now we just need to calculate \( 8 \times 7 \times 6 \times 5 \):
1. First, calculate \( 8 \times 7 = 56 \)
2. Next, calculate \( 56 \times 6 = 336 \)
3. Finally, calculate \( 336 \times 5 = 1680 \)

### Step 6: Conclusion
Thus, the value of \( \frac{8!}{4!} \) is \( 1680 \).

### Final Answer
The value of \( \frac{8!}{4!} \) is \( 1680 \).

---