What is the value of n, if P ( 15,n-1) `:` P ( 16,n-2) `= 3:4` ?

Options

A. 10
B. 12
C. 14
D. 15

Answer

To solve the problem, we need to find the value of \( n \) given the equation:

\[ \frac{P(15, n-1)}{P(16, n-2)} = \frac{3}{4} \]

### Step 1: Write the permutations in terms of factorials
Recall that the permutation \( P(n, r) \) can be expressed as:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

Using this, we can rewrite \( P(15, n-1) \) and \( P(16, n-2) \):

\[ P(15, n-1) = \frac{15!}{(15 - (n-1))!} = \frac{15!}{(16-n)!} \]

\[ P(16, n-2) = \frac{16!}{(16 - (n-2))!} = \frac{16!}{(18-n)!} \]

### Step 2: Substitute the permutations into the ratio
Now, substituting these into the ratio gives us:

\[ \frac{\frac{15!}{(16-n)!}}{\frac{16!}{(18-n)!}} = \frac{3}{4} \]

### Step 3: Simplify the ratio
This simplifies to:

\[ \frac{15! \cdot (18-n)!}{16! \cdot (16-n)!} = \frac{3}{4} \]

Since \( 16! = 16 \cdot 15! \), we can further simplify:

\[ \frac{(18-n)!}{16 \cdot (16-n)!} = \frac{3}{4} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:

\[ 4(18-n)! = 48(16-n)! \]

### Step 5: Expand the factorials
Now, we can expand \( (18-n)! \) in terms of \( (16-n)! \):

\[ (18-n)! = (18-n)(17-n)(16-n)! \]

Substituting this into the equation gives:

\[ 4(18-n)(17-n)(16-n)! = 48(16-n)! \]

### Step 6: Cancel out \( (16-n)! \)
Assuming \( (16-n)! \neq 0 \), we can cancel it out:

\[ 4(18-n)(17-n) = 48 \]

### Step 7: Simplify the equation
Dividing both sides by 4:

\[ (18-n)(17-n) = 12 \]

### Step 8: Expand and rearrange
Expanding gives:

\[ 306 - 35n + n^2 = 12 \]

Rearranging leads to:

\[ n^2 - 35n + 294 = 0 \]

### Step 9: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to 294 and add to -35. The numbers are -21 and -14:

\[ (n - 21)(n - 14) = 0 \]

### Step 10: Solve for \( n \)
Setting each factor to zero gives:

1. \( n - 21 = 0 \) → \( n = 21 \)
2. \( n - 14 = 0 \) → \( n = 14 \)

### Conclusion
The possible values for \( n \) are 14 and 21. Since the question asks for the value of \( n \), we can choose \( n = 14 \) as it is provided in the options.

Thus, the final answer is:

\[ \boxed{14} \]