Question

The sum ${}^n{C}_0{+}^n{C}_1{+}^n{C}_2+\dots \dots {+}^n{C}_n$ is equal to

• A. $\frac{2.4.6\dots ..2n}{n!}$
• B. $n^n$
• C. $n!$
• D. $3^n$

Answer 1

The correct option is A $\frac{2.4.6\dots ..2n}{n!}$

We know, $(x+a{)}^n=n_{C_0}{x}^n+n_{C_1}{x}^{n-1}a+n_{C_2}{x}^{n-2}{a}^2+....+n_{C_n}{a}^n$

Let $x=1,a=1$, then,

$2^n=n_{C_0}+n_{C_1}+n_{C_2}+....+n_{C_n}$

$n_{C_0}+n_{C_1}+n_{C_2}+....+n_{C_n}=2^n=\frac{2\cdot 1\times 2\cdot 2\times 2\cdot 3...........\times 2\cdot n}{1\times 2\times 3...........\times n}=\frac{2.4.6\dots ..2n}{n!}$