Determine n if (i) `^2n C_2:^n C_2=12 :1` (ii) `^2n C_3:^n C_3=11 :1`

Explanation

To solve the problem, we need to determine the value of \( n \) for the two given conditions. Let's break down the solution step by step.

### Part (i): \( \frac{^2n C_2}{^n C_2} = \frac{12}{1} \)

1. **Write the combinations in terms of factorials**:
\[ ^2n C_2 = \frac{(2n)!}{(2n-2)! \cdot 2!} \quad \text{and} \quad ^n C_2 = \frac{n!}{(n-2)! \cdot 2!} \]
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