Find n if :
` P(n, 6 ) = 3( P(n,5)`

Answer

To solve the equation \( P(n, 6) = 3 \cdot P(n, 5) \), we will follow these steps:

### Step 1: Write the expression for permutations
We start with the given equation:
\[ P(n, 6) = 3 \cdot P(n, 5) \]

### Step 2: Use the formula for permutations
The formula for permutations \( P(n, r) \) is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Using this, we can rewrite both sides of the equation:
\[ \frac{n!}{(n - 6)!} = 3 \cdot \frac{n!}{(n - 5)!} \]

### Step 3: Simplify the equation
Now, we can cancel \( n! \) from both sides (assuming \( n! \neq 0 \)):
\[ \frac{1}{(n - 6)!} = 3 \cdot \frac{1}{(n - 5)!} \]
This simplifies to:
\[ \frac{1}{(n - 6)!} = \frac{3}{(n - 5)!} \]

### Step 4: Express \( (n - 5)! \) in terms of \( (n - 6)! \)
Recall that:
\[ (n - 5)! = (n - 5) \cdot (n - 6)! \]
Substituting this into our equation gives:
\[ \frac{1}{(n - 6)!} = \frac{3}{(n - 5) \cdot (n - 6)!} \]

### Step 5: Cancel \( (n - 6)! \) from both sides
Assuming \( (n - 6)! \neq 0 \), we can cancel it:
\[ 1 = \frac{3}{n - 5} \]

### Step 6: Solve for \( n \)
Now, we can solve for \( n \):
\[ n - 5 = 3 \implies n = 3 + 5 = 8 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{8} \]