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

Explanation

To solve the problem, we need to find the value of \( n \) such that the ratio of permutations \( P(15, n-1) \) to \( P(16, n-2) \) equals \( \frac{3}{4} \).

### Step-by-Step Solution:

1. **Write the given ratio in terms of permutations:**
\[ \frac{P(15, n-1)}{P(16, n-2)} = \frac{3}{4} \]

2. **Express the permutations using factorials:**
\[ P(15, n-1) = \frac{15!}{(15 - (n-1))!} = \frac{15!}{(16 - n)!} \]
\[ P(16, n-2) = \frac{16!}{(16 - (n-2))!} = \frac{16!}{(18 - n)!} \]

3. **Substituting these expressions into the ratio:**
\[ \frac{\frac{15!}{(16 - n)!}}{\frac{16!}{(18 - n)!}} = \frac{3}{4} \]

4. **Simplifying the left-hand side:**
\[ \frac{15! \cdot (18 - n)!}{16! \cdot (16 - n)!} = \frac{3}{4} \]
Since \( 16! = 16 \cdot 15! \), we can simplify:
\[ \frac{(18 - n)!}{16 \cdot (16 - n)!} = \frac{3}{4} \]

5. **Cross-multiplying to eliminate the fraction:**
\[ 4(18 - n)! = 3 \cdot 16 \cdot (16 - n)! \]

6. **Rearranging the equation:**
\[ 4(18 - n)! = 48(16 - n)! \]

7. **Dividing both sides by \( (16 - n)! \):**
\[ 4 \cdot \frac{(18 - n)!}{(16 - n)!} = 48 \]
This simplifies to:
\[ 4 \cdot (18 - n)(17 - n) = 48 \]

8. **Dividing both sides by 4:**
\[ (18 - n)(17 - n) = 12 \]

9. **Expanding the left-hand side:**
\[ 306 - 35n + n^2 = 12 \]

10. **Rearranging to form a quadratic equation:**
\[ n^2 - 35n + 294 = 0 \]

11. **Factoring the quadratic equation:**
\[ (n - 14)(n - 21) = 0 \]

12. **Finding the values of \( n \):**
\[ n = 14 \quad \text{or} \quad n = 21 \]

13. **Considering the context of the problem:**
Since \( n \) must be less than or equal to 16 (as \( P(16, n-2) \) must be defined), we discard \( n = 21 \).

14. **Final answer:**
\[ n = 14 \]