Question

The sum $\underset{0<i<j\le 10}{\sum \sum }{(}^{10}{C}_j\left){(}^j{C}_i\right)$ is equal to :

• A. $2^{10}-1$
• B. $2^{10}$
• C. $3^{10}-1$
• D. $3^{10}$

Answer 1

The correct option is C $3^{10}-1$

Let $S=$ Required sum

$S={}^{10}{C}_1({}^1{C}_0+1C_1)+10C_2({}^2{C}_0+{}^2{C}_1+{}^2{C}_2)+\cdots +{}^{10}{C}_{10}({}^{10}{C}_0+{}^{10}{C}_1+cdots+{}^{10}{C}_{10})$

$\Rightarrow S={}^{10}{C}_1\left(2\right)+{}^{10}{C}_2\left(2^2\right)+{}^{10}{C}_3\left(2^3\right)+\cdots +{}^{10}{C}_{10}(2^{10}$

$\Rightarrow S=(2+1{)}^{10}-1$

$\Rightarrow S=3^{10}-1$