If `(1)/(4!) + (1)/(5!)=(x)/(6!)`, find the value of x.

Explanation

To solve the equation \(\frac{1}{4!} + \frac{1}{5!} = \frac{x}{6!}\), we will follow these steps:

### Step 1: Calculate \(4!\) and \(5!\)
First, we need to calculate the factorials:
- \(4! = 4 \times 3 \times 2 \times 1 = 24\)
- \(5! = 5 \times 4! = 5 \times 24 = 120\)

### Step 2: Rewrite the fractions
Now we can rewrite the fractions:
\[ \frac{1}{4!} = \frac{1}{24}, \quad \frac{1}{5!} = \frac{1}{120} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of \(24\) and \(120\) is \(120\). We will rewrite \(\frac{1}{24}\) with a denominator of \(120\):
\[ \frac{1}{24} = \frac{5}{120} \]

### Step 4: Add the fractions
Now we can add the two fractions:
\[ \frac{5}{120} + \frac{1}{120} = \frac{5 + 1}{120} = \frac{6}{120} \]

### Step 5: Set the equation
Now we have:
\[ \frac{6}{120} = \frac{x}{6!} \]
We know that \(6! = 720\).

### Step 6: Rewrite the equation
We can rewrite the equation:
\[ \frac{6}{120} = \frac{x}{720} \]

### Step 7: Cross-multiply
Now we cross-multiply to solve for \(x\):
\[ 6 \times 720 = 120 \times x \]
\[ 4320 = 120x \]

### Step 8: Solve for \(x\)
Now, divide both sides by \(120\):
\[ x = \frac{4320}{120} = 36 \]

### Final Answer
Thus, the value of \(x\) is \(36\).

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