Explanation

(b)
Since, a student is allowed to select at most $n$ books out of $(2n + 1)books$
If $T$ is the total number of ways selecting one book, then
$T = \ ^{2n + 1}C_{1} + \ ^{2n + 1}C_{2} + ... + \ ^{2n + 1}C_{n} = 63$ …(i)
Using the binomial theorem
$\ ^{2n + 1}C_{0} + \ ^{2n + 1}C_{1} + ... + \ ^{2n + 1}C_{n} + \ ^{2n + 1}C_{n + 1} + ...$
$= (1 + 1)^{2n + 1} = 2^{2n + 1}$
$\Rightarrow \ \ \ ^{2n + 1}C_{0} + 2\left( \ ^{2n + 1}C_{1} + \ ^{2n + 1}C_{2} + \ldots + \ ^{2n + 1}C_{n} \right) + \ ^{2n + 1}C_{2n + 1} = 2^{2n + 1}$
$\Rightarrow 1 + 2(63) + 1 = 2^{2n + 1}$
$\Rightarrow \ \ \ \ 1 + 63 = \frac{2^{2n + 1}}{2} = 2^{n}$
$\Rightarrow \ \ \ 2^{6} = 2^{2n}\ \Rightarrow \ \ \ n = 3$