If `(1)/(lfloor8) + (1)/(lfloor9) = (x)/(lfloor10)` , then x is equal to :

Options

A. 10
B. 20
C. 9
D. 100

Answer

To solve the equation \(\frac{1}{\lfloor 8! \rfloor} + \frac{1}{\lfloor 9! \rfloor} = \frac{x}{\lfloor 10! \rfloor}\), we first need to understand the factorial notation and then manipulate the equation to find the value of \(x\).

### Step-by-step Solution:

1. **Understand Factorials**:
- The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\).
- For example:
- \(8! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8\)
- \(9! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9\)
- \(10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10\)

2. **Rewrite the Equation**:
- The equation can be rewritten as:
\[ \frac{1}{8!} + \frac{1}{9!} = \frac{x}{10!} \]

3. **Find a Common Denominator**:
- The common denominator for the left side is \(9!\):
\[ \frac{9!}{8! \cdot 9!} + \frac{8!}{9! \cdot 9!} = \frac{9! + 1}{9!} \]
- Therefore, we can rewrite the left side as:
\[ \frac{9! + 1}{9!} \]

4. **Express \(10!\) in Terms of \(9!\)**:
- We know that \(10! = 10 \times 9!\).

5. **Substituting Back into the Equation**:
- Now, substituting back into the equation gives:
\[ \frac{9! + 1}{9!} = \frac{x}{10 \times 9!} \]

6. **Cross-Multiply**:
- Cross-multiplying gives:
\[ (9! + 1) \cdot 10 = x \]

7. **Calculate \(x\)**:
- We can simplify this to:
\[ x = 10 \cdot 9! + 10 \]
- Since \(9! = 362880\), we can calculate:
\[ x = 10 \cdot 362880 + 10 = 3628800 + 10 = 3628810 \]

8. **Final Value of \(x\)**:
- Therefore, the value of \(x\) is:
\[ x = 3628810 \]