Question

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

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $3$

The no. of ways he can select at most n books from a collection of (2n + 1) books is :

${}^{2n+1}{C}_0$ + ${}^{2n+1}{C}_1$ + ....................${+}^{2n+1}{C}_n$ = $S$

We know that sum of all coefficients of form ${}^n{C}_r$ adds up to $2^n$ and ${}^n{C}_r{=}^n{C}_{n-r}$

$S=2^{(2n+1)-1}$

$=2^{2n}$

But the student must choose at least one book he cannot choose 0.

So $S{-}^{2n+1}{C}_0=63$

$\Rightarrow S=64$

$2^{2n}=64$

Hence

$n=3$