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 at least one book is $63$, find the value of $n$.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

Given,${}_1^{2n+1}\text{C}{+}_2^{2n+1}\text{C}{+}_3^{2n+1}\text{C}+.....{+}_n^{2n+1}\text{C}=63=S$
we know,
${}_0^{2n+1}\text{C}{+}_1^{2n+1}\text{C}+....{+}_{2n}^{2n+1}\text{C}{+}_{2n+1}^{2n+1}\text{C}=2^{2n+1}$
${}_0^{2n+1}\text{C}{=}_{2n+1}^{2n+1}\text{C}=1$
and ${}_1^{2n+1}\text{C}{=}_{2n}^{2n+1}\text{C}....$
so,${}_0^{2n+1}\text{C}+2S{+}_{2n+1}^{2n+1}\text{C}=2^{2n+1}$
$2+2S=2^{2n+1}$
$2+126=2^{2n+1}=128=2^7$
$=>2n+1=7$
$2n=6=>n=3$