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If $C_{r}$ denotes the binomial coefficient $^{n} C_{r}$ then $(- 1) C_{0}^{2} + 2 C_{1}^{2} + 5 C_{2}^{2} + \dots + (3 n - 1) C_{n}^{2} =$

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${(3 n - 2)}^{2 n} C_{n}$

${(5 + 3 n)}^{2 n} C_{n}$

${(\frac{3 n - 2}{2})}^{2 n} C_{n}$

${(\frac{3 n - 5}{2})}^{2 n} C_{n + 1}$

answer is B.

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Detailed Solution

$(- 1) C_{0}^{2} + 2 \cdot C_{1}^{2} + 5 \cdot C_{2}^{2} + \dots + (3 n - 1) \cdot C_{n}^{2} = \frac{- 1 + 3 n - 1}{2} (C_{0}^{2} + C_{1}^{2} + \dots + C_{n}^{2}) = {\frac{3 n - 2}{2}}^{2 n} C_{n}$

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