If `1/(1!11!)+1/(3!9!)+1/(5!7!)=(2^n)/(m!)` then the value of m + n is

Options

A. 18
B. 23
C. 12
D. 22

Answer

To solve the equation

\[ \frac{1}{1! \cdot 11!} + \frac{1}{3! \cdot 9!} + \frac{1}{5! \cdot 7!} = \frac{2^n}{m!} \]

we will simplify the left-hand side step by step.

### Step 1: Write the left-hand side clearly
The left-hand side (LHS) is:

\[ LHS = \frac{1}{1! \cdot 11!} + \frac{1}{3! \cdot 9!} + \frac{1}{5! \cdot 7!} \]

### Step 2: Multiply and divide by \(12!\)
To simplify the expression, we multiply and divide each term by \(12!\):

\[ LHS = \frac{12!}{1! \cdot 11! \cdot 12!} + \frac{12!}{3! \cdot 9! \cdot 12!} + \frac{12!}{5! \cdot 7! \cdot 12!} \]

This gives us:

\[ LHS = \frac{12!}{1! \cdot 11!} + \frac{12!}{3! \cdot 9!} + \frac{12!}{5! \cdot 7!} \]

### Step 3: Simplify each term
Now we simplify each term separately:

1. For the first term:
\[ \frac{12!}{1! \cdot 11!} = \frac{12 \cdot 11!}{1! \cdot 11!} = 12 \]

2. For the second term:
\[ \frac{12!}{3! \cdot 9!} = \frac{12 \cdot 11 \cdot 10 \cdot 9!}{3! \cdot 9!} = \frac{12 \cdot 11 \cdot 10}{6} = 2 \cdot 11 \cdot 10 = 220 \]

3. For the third term:
\[ \frac{12!}{5! \cdot 7!} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!}{5! \cdot 7!} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{120} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5!} \]
Simplifying gives:
\[ = 792 \]

### Step 4: Add the simplified terms
Now we add these simplified terms together:

\[ LHS = 12 + 220 + 792 = 1024 \]

### Step 5: Write the final expression
Now we have:

\[ LHS = \frac{1024}{12!} \]

### Step 6: Compare with the right-hand side
We know from the problem statement that:

\[ \frac{1024}{12!} = \frac{2^n}{m!} \]

Since \(1024 = 2^{10}\), we can write:

\[ \frac{2^{10}}{12!} \]

From this, we can identify \(n = 10\) and \(m = 12\).

### Step 7: Calculate \(m + n\)
Finally, we calculate:

\[ m + n = 12 + 10 = 22 \]

Thus, the final answer is:

\[ \boxed{22} \]