Question

A student is allowed to select almost $n$ books from a collection of $2n+1$ books. If the total number of ways in which he can select at least one book is $63$, find the value of $n$.

Answer 1

Number of ways in which a student can select at least one and almost $n$ books out of $2n+1$ books is

${}^{2n+1}{C}_1{+}^{2n+1}{C}_2{+}^{2n+1}{C}_3+...{+}^{2n+1}{C}_n$

$=\frac{1}{2}[2{\times }^{2n+1}{C}_1+2{\times }^{2n+1}{C}_2+2{\times }^{2n+1}{C}_3+...+2{\times }^{2n+1}{C}_n]$

$=\frac{1}{2}\left[{(}^{2n+1}{C}_1{+}^{2n+1}{C}_{2n}\right)+{(}^{2n+1}{C}_2{+}^{2n+1}{C}_{2n-1})+{(}^{2n+1}{C}_3{+}^{2n+1}{C}_{2n-1})+...+{(}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}\left)\right]$

[Using ${}^n{C}_r{=}^n{C}_{n-r}]$

$\frac{1}{2}{[}^{2n+1}{C}_1{+}^{2n+1}{C}_2{+}^{2n+1}{C}_3+...{+}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}{+}^{2n+1}{C}_{n+2}+...{+}^{2n+1}{C}_{2n}]$

$=\frac{1}{2}{[}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+...{+}^{2n+1}{C}_{2n+1}-1-1]$

$=\frac{1}{2}[2^{2n+1}-1]=2^{2n}-1$

Now given,
$2^{2n}-1=63$

$\Rightarrow 2^{2n}=64=2^6$

$\Rightarrow 2n=6$

$\Rightarrow n=3$.