Explanation

(c)
Given that,
${(1 + x)}^{n} = \ ^{n}C_{0} + \ ^{n}C_{1}x + \ ^{n}C_{2}x^{2} + ... + \ ^{n}C_{n}x^{n}$
Let $S_{n} = \frac{\ ^{n}C_{1}}{\ ^{n}C_{0}} + \frac{2\ ^{n}C_{2}}{\ ^{n}C_{1}} + \frac{3\ ^{n}C_{3}}{\ ^{n}C_{2}} + ... + \frac{n\ ^{n}C_{n}}{\ ^{n}C_{n - 1}}$
Put $n = 1,2,3,\ldots$, then
$S_{1} = \frac{\ ^{1}C_{1}}{\ ^{1}C_{0}} = 1$,
$S_{2} = \frac{\ ^{2}C_{1}}{\ ^{2}C_{0}} + 2\frac{\ ^{2}C_{2}}{\ ^{2}C_{1}}$
$= \frac{2}{1} + 2 \bullet \frac{1}{2} = 2 + 1 = 3$
By taking option, (put $n = 1,2,\ldots$) (a) and (b) does not hold condition, but option (c) satisfies.