Question

If \( \frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!} \), then the value of \( x \) is:

• A. 121
• B. 120
• C. 12
• D. 24

Hint:

In factorial equations, look for ways to factor and cancel terms to simplify the equation.

Answer 1

The Correct Option is B

Step 1: Write the equation.
The given equation is: \[ \frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!} \]

Step 2: Simplify the terms.
We can factor \( \frac{1}{9!} \) out of the left-hand side of the equation: \[ \frac{1}{9!} \left( 1 + \frac{1}{10} \right) = \frac{x}{11!} \] \[ \frac{1}{9!} \left( \frac{10 + 1}{10} \right) = \frac{x}{11!} \] \[ \frac{1}{9!} \times \frac{11}{10} = \frac{x}{11!} \]

Step 3: Further simplify.
Now, multiply both sides by \( 11! \) to isolate \( x \): \[ \frac{11!}{9! \times 10} = x \] \[ x = \frac{11 \times 10!}{10} = 120 \] Thus, the value of \( x \) is 120. Therefore, the correct answer is 2. 120.