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} $