Find the value of n if `(n)/(11!) = (1)/(9!) +(1)/(10!)`.

Answer

To solve the equation \( \frac{n}{11!} = \frac{1}{9!} + \frac{1}{10!} \), we will follow these steps:

### Step 1: Rewrite the right-hand side
We start with the right-hand side of the equation:
\[ \frac{1}{9!} + \frac{1}{10!} \]
We can express \( \frac{1}{10!} \) in terms of \( 9! \):
\[ \frac{1}{10!} = \frac{1}{10 \times 9!} = \frac{1}{10} \cdot \frac{1}{9!} \]
Thus, we can rewrite the right-hand side as:
\[ \frac{1}{9!} + \frac{1}{10!} = \frac{1}{9!} + \frac{1}{10} \cdot \frac{1}{9!} = \frac{1 + \frac{1}{10}}{9!} = \frac{10 + 1}{10 \cdot 9!} = \frac{11}{10 \cdot 9!} \]

### Step 2: Set the equation
Now we can set the equation:
\[ \frac{n}{11!} = \frac{11}{10 \cdot 9!} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ n \cdot 10 \cdot 9! = 11 \cdot 11! \]

### Step 4: Simplify \( 11! \)
We know that \( 11! = 11 \times 10 \times 9! \). Substituting this into our equation:
\[ n \cdot 10 \cdot 9! = 11 \cdot (11 \times 10 \times 9!) \]
We can cancel \( 9! \) from both sides (assuming \( 9! \neq 0 \)):
\[ n \cdot 10 = 11 \cdot 11 \cdot 10 \]

### Step 5: Solve for \( n \)
Now we can simplify:
\[ n \cdot 10 = 1210 \]
Dividing both sides by 10:
\[ n = 121 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{121} \]

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