Question

If a student can select maximum $n$ books (and at least $1$ book) from $2n+1$ books in $63$ ways. Then value of $n$ is

• A. $3$
• B. $5$
• C. $7$
• D. $9$

Answer 1

The correct option is A $3$

Selecting maximum $n$ object from $2n+1$ objects
${}^{2n+1}{C}_1+{}^{2n+1}{C}_2+\cdots +{}^{2n+1}{C}_n=63\Rightarrow {}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+\cdots +{}^{2n+1}{C}_n=63+1\Rightarrow 2[{}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+\cdots +{}^{2n+1}{C}_n]=128\Rightarrow {}^{2n+1}{C}_0+{}^{2n+1}{C}_1+\cdots +{}^{2n+1}{C}_n+{}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_{n+2}+\cdots ++{}^{2n+1}{C}_{2n+1}=2^7\Rightarrow 2^{2n+1}=2^7\Rightarrow n=3$