Question

Total number of books is $2n + 1$. One is allowed to select a minimum of the one book and a maximum of $n$ books. If total number of selections if $63$, then value of $n$ is :

• A. 3
• B. 6
• C. 2
• D. none of these

Answer 1

The Correct Option is A

Since $(1+x)^{2 n+1}=C_{0}+C_{1} x+\ldots+ C_{n} x^{n}$ $+C_{n+1} x^{n+1}+\ldots \ldots+ x^{2 n+1}$ $=2\left(C_{0}+C_{1}+\ldots . C_{n} x^{2}\right)$ put $x=1 $ $(1+1)^{2 n+1}=2\left(C_{0}+C_{1}+\ldots \ldots+C_{n}\right)$ $\Rightarrow 2^{2 n}=\left(c_{0}+C_{1}+C_{2}+\ldots+C_{n}\right.$ $\Rightarrow 2^{2 n}-1=63$ $\Rightarrow 2^{2 n}=64 \Rightarrow 2^{2 n}=2^{6}$ $\Rightarrow 2 n=6 \Rightarrow n=3$