Question

The total number of selections of at most n things from $(2n+1)$ different things is 63. Then the value of n is

• A. 3
• B. 2
• C. 4
• D. none of these

Answer 1

The correct option is A 3

The total number of selections of at most n things from (2n+1) different things i.e. selecting 1,2,3,......n things from (2n+1) things
$⟹$ ${}^{2n+1}{C}_1$+${}^{2n+1}{C}_2$+${}^{2n+1}{C}_3$+............................${}^{2n+1}{C}_n$
$⟹$ $\frac{1}{2}{(}^{2n+1}{C}_1$+${}^{2n+1}{C}_2$+${}^{2n+1}{C}_3$+............................${}^{2n+1}{C}_{2n})$
$⟹$ $\frac{1}{2}{(}^{2n+1}{C}_0{+}^{2n+1}{C}_1$+${}^{2n+1}{C}_2$+${}^{2n+1}{C}_3$+............................${}^{2n+1}{C}_{2n}{+}^{2n+1}{C}_{2n+1})$-1
$⟹$$\frac{1}{2}\left(2^{2n+1}\right)$-1
$⟹$$2^{2n}-1$=63
$⟹$$2^{2n}=2^6$
$\Rightarrow n=3$