A) \[\frac{{{2}^{50}}}{51}\]
B) \[\frac{{{2}^{50}}-1}{51}\]
C) \[\frac{{{2}^{50}}-1}{50}\]
D) \[\frac{{{2}^{51}}-1}{51}\]
E) \[\frac{{{2}^{51}}-1}{50}\]
Correct Answer: A
Solution :
\[\left( \frac{^{50}{{C}_{0}}}{1}+\frac{^{50}{{C}_{2}}}{3}+\frac{^{50}{{C}_{4}}}{5}+....+\frac{^{50}{{C}_{50}}}{51} \right)\] \[=\frac{1}{1}+\frac{50\times 49}{3\times 2!}+\frac{50\times 49\times 48\times 47}{5\times 4!}+....\] \[=\frac{1}{51}\left( 51+\frac{51\times 50\times 49}{3!}+\frac{\begin{align} & 51\times 50\times 49 \\ & \times 48\times 47 \\ \end{align}}{5!}+.... \right)\] \[=\frac{1}{51}{{(}^{51}}{{C}_{1}}{{+}^{51}}{{C}_{3}}{{+}^{51}}{{C}_{5}}+....)\] \[=\frac{1}{51}{{.2}^{51-1}}=\frac{{{2}^{50}}}{51}\]
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