(i) If `""^(2n)C_(3) : ""^(n)C_(3)= 11 : 1`, find n.
(ii) If `""^(2n)C_(3) : ""^(n)C_(2) = 12 : 1`, find n.

Answer

To solve the given problems step by step, we will use the formula for combinations, which is given by:

\[ ^nC_r = \frac{n!}{r!(n-r)!} \]

### Part (i): If \(^{{2n}}C_3 : ^nC_3 = 11 : 1\), find \(n\).

1. **Write the ratio using the combination formula**:
\[ \frac{^{{2n}}C_3}{^nC_3} = \frac{\frac{(2n)!}{3!(2n-3)!}}{\frac{n!}{3!(n-3)!}} = \frac{(2n)!}{(2n-3)!} \cdot \frac{(n-3)!}{n!} \]

2. **Simplify the expression**:
\[ = \frac{(2n)(2n-1)(2n-2)}{n(n-1)(n-2)} \]

3. **Set the ratio equal to 11**:
\[ \frac{(2n)(2n-1)(2n-2)}{n(n-1)(n-2)} = 11 \]

4. **Cross-multiply**:
\[ (2n)(2n-1)(2n-2) = 11n(n-1)(n-2) \]

5. **Expand both sides**:
- Left side:
\[ 2n(2n-1)(2n-2) = 2n(4n^2 - 6n + 2) = 8n^3 - 12n^2 + 4n \]
- Right side:
\[ 11n(n-1)(n-2) = 11(n^3 - 3n^2 + 2n) = 11n^3 - 33n^2 + 22n \]

6. **Set the equation**:
\[ 8n^3 - 12n^2 + 4n = 11n^3 - 33n^2 + 22n \]

7. **Rearranging the equation**:
\[ 0 = 3n^3 - 21n^2 + 18n \]

8. **Factor out common terms**:
\[ 0 = 3n(n^2 - 7n + 6) \]

9. **Factoring the quadratic**:
\[ n^2 - 7n + 6 = (n - 1)(n - 6) = 0 \]

10. **Find the values of \(n\)**:
\[ n = 1 \quad \text{or} \quad n = 6 \]

### Part (ii): If \(^{{2n}}C_3 : ^nC_2 = 12 : 1\), find \(n\).

1. **Write the ratio using the combination formula**:
\[ \frac{^{{2n}}C_3}{^nC_2} = \frac{\frac{(2n)!}{3!(2n-3)!}}{\frac{n!}{2!(n-2)!}} = \frac{(2n)!}{(2n-3)!} \cdot \frac{2!(n-2)!}{n!} \]

2. **Simplify the expression**:
\[ = \frac{(2n)(2n-1)(2n-2)}{n(n-1)} \cdot 2 \]

3. **Set the ratio equal to 12**:
\[ \frac{2(2n)(2n-1)(2n-2)}{n(n-1)} = 12 \]

4. **Cross-multiply**:
\[ 2(2n)(2n-1)(2n-2) = 12n(n-1) \]

5. **Simplify**:
\[ (2n)(2n-1)(2n-2) = 6n(n-1) \]

6. **Expand both sides**:
- Left side:
\[ 2n(4n^2 - 6n + 2) = 8n^3 - 12n^2 + 4n \]
- Right side:
\[ 6n^2 - 6n \]

7. **Set the equation**:
\[ 8n^3 - 12n^2 + 4n = 6n^2 - 6n \]

8. **Rearranging the equation**:
\[ 8n^3 - 18n^2 + 10n = 0 \]

9. **Factor out common terms**:
\[ 2n(4n^2 - 9n + 5) = 0 \]

10. **Factoring the quadratic**:
\[ 4n^2 - 9n + 5 = (4n - 5)(n - 1) = 0 \]

11. **Find the values of \(n\)**:
\[ n = 0.25 \quad \text{or} \quad n = 1 \]

### Final Answers:
- For part (i): \(n = 1\) or \(n = 6\)
- For part (ii): \(n = 1\) or \(n = 0.25\)