# A student is allowed to select at the most n books from a collection of $2n+1$ books. The no. of ways in which he can select at least one book is 63, then $n$ = ?

$\begin{array}{1 1} 1 \\ 2 \\ 3 \\ 4 \end{array}$

The no. of ways in which one can select at least one book
and at the most $n$ books from $2n+1$ books is
$^{2n+1}C_1+^{2n+1}C_2+^{2n+1}C_3+...........^{2n+1}C_n$
we know
$(1+x)^{2n+1}=^{2n+1}C_0+^{2n+1}C_1.x+........^{2n+1}C_{2n+1}.x^n$
Put $x=1$ and also we know that $^nC_r=^nC_{n-r}$
$^{2n+1}C_0+^{2n+1}C_1+....^{2n+1}C_n+^{2n+1}C_{n+1}....^{2n+1}C_{2n+1}$
$=2^{2n+1}$
$i.e.,\:\: 2\big[^{2n+1}C_0+^{2n+1}C_1+....^{2n+1}C_n\big]=2^{2n+1}$
$\Rightarrow\:^{2n+1}C_0+^{2n+1}C_1+....^{2n+1}C_n=2^{2n}$
$\Rightarrow\:^{2n+1}C_1+....^{2n+1}C_n=2^{2n}-1$ $(Since\:^{2n+1}C_0=1)$
Given that $^{2n+1}C_1+^{2n+1}C_2+^{2n+1}C_3+...........^{2n+1}C_n=63$
$\Rightarrow\:2^{2n}-1=63$
$\Rightarrow\:2^{2n}=64$
$\Rightarrow\:n=3$
answered Oct 28, 2013