If `n! - n = n`, then the value of n is :

Options

A. 4
B. 5
C. 6
D. 3

Answer

To solve the equation \( n! - n = n \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ n! - n = n \]
We can simplify this by adding \( n \) to both sides:
\[ n! = n + n \]
This simplifies to:
\[ n! = 2n \]

### Step 2: Test integer values for \( n \)
Now, we will test integer values for \( n \) to find which one satisfies the equation \( n! = 2n \).

1. **For \( n = 1 \)**:
\[ 1! = 1 \quad \text{and} \quad 2 \times 1 = 2 \quad \Rightarrow \quad 1 \neq 2 \]

2. **For \( n = 2 \)**:
\[ 2! = 2 \quad \text{and} \quad 2 \times 2 = 4 \quad \Rightarrow \quad 2 \neq 4 \]

3. **For \( n = 3 \)**:
\[ 3! = 6 \quad \text{and} \quad 2 \times 3 = 6 \quad \Rightarrow \quad 6 = 6 \]
This satisfies the equation.

4. **For \( n = 4 \)**:
\[ 4! = 24 \quad \text{and} \quad 2 \times 4 = 8 \quad \Rightarrow \quad 24 \neq 8 \]

5. **For \( n = 5 \)**:
\[ 5! = 120 \quad \text{and} \quad 2 \times 5 = 10 \quad \Rightarrow \quad 120 \neq 10 \]

6. **For \( n = 6 \)**:
\[ 6! = 720 \quad \text{and} \quad 2 \times 6 = 12 \quad \Rightarrow \quad 720 \neq 12 \]

### Conclusion
The only integer value of \( n \) that satisfies the equation \( n! = 2n \) is:
\[ \boxed{3} \]