The value of `8! div 5!` is :

Options

A. 4
B. 16
C. 2
D. 336

Answer

To solve the question "The value of `8! div 5!`", we will follow these steps:

### Step 1: Write down the factorials
We start by expressing the factorials:
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

### Step 2: Set up the division
Now we can set up the division of these two factorials:
\[ \frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \]

### Step 3: Cancel common terms
Notice that \( 5! \) (which is \( 5 \times 4 \times 3 \times 2 \times 1 \)) appears in both the numerator and the denominator. We can cancel these terms:
\[ \frac{8 \times 7 \times 6 \times \cancel{5 \times 4 \times 3 \times 2 \times 1}}{\cancel{5 \times 4 \times 3 \times 2 \times 1}} = 8 \times 7 \times 6 \]

### Step 4: Calculate the remaining product
Now we need to calculate \( 8 \times 7 \times 6 \):
1. First, calculate \( 8 \times 7 = 56 \).
2. Next, multiply the result by 6: \( 56 \times 6 = 336 \).

### Step 5: Final result
Thus, the value of \( \frac{8!}{5!} \) is \( 336 \).

### Conclusion
The final answer is:
\[ \frac{8!}{5!} = 336 \]