Question

A student is allowed to select at the most $n$ books from a collection of $(2n+1)$ books.

If the total number of ways in which he can select the books is $63$, find the value of $n$.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

Given, a student selects at the most $n$ books from a collection of $(2n+1)$ books.

It means that he selects one book or two books or three books or ... or n books. Hence, by the given hypothesis
${}^{2n+1}{C}_1{+}^{2n+1}{C}_2{+}^{2n+1}{C}_3+...{+}^{2n+1}{C}_n=63$
But we know that
${}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2{+}^{2n+1}{C}_3+...{+}^{2n+1}{C}_{2n+1}=2^{2n+1}$
Now,
${}^{2n+1}{C}_0{=}^{2n+1}{C}_{2n+1}=1$

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

$\Rightarrow 2+2.63=2^{2n+1}$
$\Rightarrow 1+63=2^{2n}$
$\Rightarrow 64=2^{2n}$
$\Rightarrow 2^6=2^{2n}$
$\Rightarrow 2n=6$
$\therefore n=3$