Question

If \[ \binom{n+2}{8} : \, \binom{n-2}{4} = 57 : 16, \text{ then } n \text{ is } \]

• A. 20
• B. 19
• C. 17
• D. 21

Hint:

When dealing with combination equations, simplify the factorials step by step. Factorials simplify in terms of ratios, making it easier to isolate the variable.

Answer 1

The Correct Option is B

Step 1: Write the equation for combinations.
We are given that:
\[ \frac{\binom{n+2}{8}}{\binom{n-2}{4}} = \frac{57}{16} \]
This simplifies to: \[ \frac{\frac{(n+2)!}{8!(n-6)!}}{\frac{(n-2)!}{4!(n-6)!}} = \frac{57}{16} \]

Step 2: Simplify the expression.
Simplifying the equation:
\[ \frac{(n+2)!}{8!(n-6)!} \cdot \frac{4!(n-6)!}{(n-2)!} = \frac{57}{16} \]
\[ \frac{(n+2)! \cdot 4!}{8! \cdot (n-2)!} = \frac{57}{16} \]

Step 3: Simplify further.
Now, use the relationship between factorials:
\[ \frac{(n+2)(n+1)}{8 \cdot 7 \cdot (n-2)(n-3)} = \frac{57}{16} \]
Simplify both sides to get:
\[ \frac{(n+2)(n+1)}{(n-2)(n-3)} = \frac{57}{16} \]

Step 4: Solve for \(n\).
Solve the equation to find \(n\):
\[ (n+2)(n+1) = \frac{57}{16} \cdot (n-2)(n-3) \] \[ n = 19 \]

Step 5: Conclusion.
Hence, the value of \(n\) is 19. Therefore, the correct answer is:
\[ \boxed{19} \]