Explanation

(a)
$\left( \frac{\ ^{50}C_{0}}{1} + \frac{\ ^{50}C_{2}}{3} + \frac{\ ^{50}C_{4}}{5} + \ldots + \frac{\ ^{50}C_{50}}{51} \right)$
$= \frac{1}{1} + \frac{50 \times 49}{3 \times 2! \times} + \frac{50 \times 49 \times 48 \times 47}{5 \times 4!} + \ldots$
$= \frac{1}{51}\left( 51 + \frac{51 \times 50 \times 49}{3!} + \frac{51 \times 50 \times 49 \times 48 \times 47}{5!} + \ldots \right)$
$= \frac{1}{51}\left( \ ^{51}C_{1} + \ ^{51}C_{3} + \ ^{51}C_{5} + \ldots \right) = \frac{1}{51} \bullet 2^{51 - 1} = \frac{2^{50}}{51}$