If `P(n,6) = 30 P(n,4)`, then `n = ?`

Options

A. 6
B. 9
C. 8
D. None of these

Answer

To solve the equation \( P(n, 6) = 30 P(n, 4) \), we can follow these steps:

### Step 1: Write the Permutation Formula
The permutation \( P(n, r) \) is given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
For our problem, we have:
\[ P(n, 6) = \frac{n!}{(n - 6)!} \quad \text{and} \quad P(n, 4) = \frac{n!}{(n - 4)!} \]

### Step 2: Substitute the Permutation Values
Substituting these into the equation \( P(n, 6) = 30 P(n, 4) \), we get:
\[ \frac{n!}{(n - 6)!} = 30 \cdot \frac{n!}{(n - 4)!} \]

### Step 3: Cancel \( n! \) from Both Sides
Since \( n! \) is common on both sides, we can cancel it out (assuming \( n! \neq 0 \)):
\[ \frac{1}{(n - 6)!} = 30 \cdot \frac{1}{(n - 4)!} \]

### Step 4: Rewrite \( (n - 6)! \)
We can express \( (n - 4)! \) in terms of \( (n - 6)! \):
\[ (n - 4)! = (n - 4)(n - 5)(n - 6)! \]
Thus, we can rewrite the equation as:
\[ \frac{1}{(n - 6)!} = \frac{30}{(n - 4)(n - 5)(n - 6)!} \]

### Step 5: Multiply Both Sides by \( (n - 6)! \)
Multiplying both sides by \( (n - 6)! \) gives us:
\[ 1 = \frac{30}{(n - 4)(n - 5)} \]

### Step 6: Cross-Multiply
Cross-multiplying results in:
\[ (n - 4)(n - 5) = 30 \]

### Step 7: Expand the Equation
Expanding the left side:
\[ n^2 - 9n + 20 = 30 \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ n^2 - 9n - 10 = 0 \]

### Step 9: Factor the Quadratic Equation
Factoring the quadratic:
\[ (n - 10)(n + 1) = 0 \]

### Step 10: Solve for \( n \)
Setting each factor to zero gives:
\[ n - 10 = 0 \quad \Rightarrow \quad n = 10 \]
\[ n + 1 = 0 \quad \Rightarrow \quad n = -1 \]

### Step 11: Determine Valid Solutions
Since \( n \) must be a positive integer (as it represents the number of items), we discard \( n = -1 \) and accept:
\[ n = 10 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{10} \]