Question

The value of $\left(\frac{{}^{50}{C}_0}{1}+\frac{{}^{50}{C}_2}{3}+.....+\frac{{}^{50}{C}_{50}}{51}\right)$ is

• A. $\frac{2^{50}}{51}$
• B. (b) $\frac{2^{50}-1}{51}$
• C. $\frac{2^{50}-1}{50}$
• D. None of these

Answer 1

Answer: (A)

$\left(\frac{{}^{50}{C}_0}{1}+\frac{{}^{50}{C}_2}{3}+......+\frac{{}^{50}{C}_{50}}{51}\right)$ $=\left(\frac{1}{1}+\frac{50\times 49}{3\times 2!}+....+\frac{1}{51}\right)$ $=\frac{1}{51}\left(\frac{51}{1!}+\frac{51\times 50\times 49}{3!}+.....+1\right)$ $=\frac{1}{51}({}^{51}{C}_1{+}^{51}{C}_3+...{+}^{51}{C}_{51})$ $=\frac{1}{51}{2}^{51-1}=\frac{2^{50}}{51}$