Question

If \[ {}^{2n}C_3 : {}^{n}C_3 = 12:1, \] then \(n=\)

• A. \(5\)
• B. \(8\)
• C. \(10\)
• D. \(3\)

Hint:

When ratios of combinations are given, first write combinations in factorial form and simplify by cancelling common factors before solving.

Answer 1

The Correct Option is A

Step 1: Write the given ratio as an equation.
Given $^{2n}C_3 : ^nC_3 = 12:1$, so $\frac{^{2n}C_3}{^nC_3} = 12$.

Step 2: Write out the combination formula.
$^{2n}C_3 = \frac{(2n)(2n-1)(2n-2)}{6}$ and $^nC_3 = \frac{n(n-1)(n-2)}{6}$.

Step 3: Form the ratio and simplify.
\[ \frac{(2n)(2n-1)(2n-2)}{n(n-1)(n-2)} = 12 \] Since $2n = 2 \cdot n$ and $2n-2 = 2(n-1)$: \[ \frac{2n \cdot (2n-1) \cdot 2(n-1)}{n(n-1)(n-2)} = 12 \implies \frac{4(2n-1)}{n-2} = 12 \]

Step 4: Solve for $n$.
$4(2n-1) = 12(n-2) \Rightarrow 8n-4 = 12n-24 \Rightarrow 20 = 4n \Rightarrow n=5$.

Step 5: Verify.
$^{10}C_3 = 120$ and $^5C_3 = 10$. Ratio $= 12$. Correct.

Step 6: State the answer.
\[ \boxed{5} \]