Question

A student is allowed to select at most $n$ books out of $(2n+1)$ books. If the total number of ways in which he can select a book is $63$, find the value of $n$.

• A. $6$
• B. $3$
• C. $4$
• D. $2$

Answer 1

Answer: (B)

$3$

Almost $n$ Books out of $(2n+1)$ books

Total Number of Ways $=63$

$n=?$

${}^{2n+1}{C}_1{+}^{2n+1}{C}_2{+}^{2n+1}{C}_3+..........{+}^{2n+1}{C}_n=(1+x{)}^{2n+1}$

$x=1{,}^n{C}_r{=}^n{C}_{n-r}$

$\Rightarrow 2\times [{}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+..........{+}^{2n+1}{C}_n]=2^{2n+1}$

$2^{2n}-1=632^{2n}=644^n={\left(4\right)}^{3}n=3$