This problem requires us to solve a mathematical equation involving factorials to find the value of x.

Solving the Factorial Equation for x

We are given the equation:

$ \frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!} $

Our goal is to find the value of x.

Understanding Factorials

First, let's recall the definition of a factorial. For any positive integer n, the factorial, denoted by n!, is the product of all positive integers up to n.

$ n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1 $

We can express larger factorials in terms of smaller ones. This is key to simplifying the equation:

• $ 10! = 10 \times 9! $
• $ 11! = 11 \times 10 \times 9! $

Simplifying the Equation Step-by-Step

Let's substitute these factorial relationships back into the original equation:

$ \frac{1}{9!} + \frac{1}{10 \times 9!} = \frac{x}{11 \times 10 \times 9!} $

To combine the terms on the left side of the equation, we need a common denominator, which is $10 \times 9!$. We rewrite the first term:

$ \frac{1 \times 10}{9! \times 10} + \frac{1}{10 \times 9!} = \frac{x}{11 \times 10 \times 9!} $

Now, add the fractions on the left side:

$ \frac{10}{10 \times 9!} + \frac{1}{10 \times 9!} = \frac{x}{11 \times 10 \times 9!} $

$ \frac{10 + 1}{10 \times 9!} = \frac{x}{11 \times 10 \times 9!} $

$ \frac{11}{10 \times 9!} = \frac{x}{11 \times 10 \times 9!} $

Notice that $10 \times 9!$ is equal to $10!$, and $11 \times 10 \times 9!$ is equal to $11!$. So the equation is:

$ \frac{11}{10!} = \frac{x}{11!} $

Calculating the Value of x

To solve for x, we can isolate it by multiplying both sides of the equation by $11!$:

$ x = \frac{11}{10!} \times 11! $

Now, substitute $11! = 11 \times 10!$:

$ x = \frac{11}{10!} \times (11 \times 10!) $

The $10!$ terms cancel out:

$ x = 11 \times 11 $

Finally, perform the multiplication:

$ x = 121 $

Conclusion

Therefore, the value of x in the given factorial equation is 121.