Explanation

(a)
We know,
${(1 + x)}^{n} = C_{0} + C_{1}x + C_{2}x^{2} + ... + C_{r}x^{r} + ...$ …(i)
and $\left( 1 + \frac{1}{x} \right)^{n} = C_{0} + C_{1}\frac{1}{x} + C_{2}\frac{1}{x^{2}} + ... + C_{r}\frac{1}{x^{r}}$
$+ C_{r + 1}\frac{1}{x^{r + 1}} + C_{r + 2}\frac{1}{x^{r + 2}}...C_{n}\frac{1}{x^{n}}$ …(ii)
On multiplying Eqs. (i) and (ii), equation coefficient of $x^{r}$ in $\frac{1}{x^{n}}{(1 + x)}^{2n}$ or the coefficient of $x^{n + r}$ in ${(1 + x)}^{2n}$, we get the value of required expression which is
$\ ^{2n}C_{n + r} = \frac{(2n)!}{(n - r)!(n + r)!}$