Question

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

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

Answer 1

Answer: (D)

$$\begin{gathered} {}^{10}{C}_0\left({}^0{C}_0\right)+{}^{10}{C}_1\left({}^1{C}_0+{}^1{C}_1\right)+{}^{10}{C}_2\left({}^2{C}_0+{}^2{C}_1+{}^2{C}_2\right)+\dots \dots .. \\ +{}^{10}{C}_{10}\left({}^{10}{C}_0+{}^{10}{C}_1+{}^{10}{C}_2+\dots \dots .+{}^{10}{C}_{10}\right) \end{gathered}$$
$={}^{10}{C}_0+{}^{10}{C}_1\cdot 2+{}^{10}{C}_2\cdot 2^2+{}^{10}{C}_3\cdot 2^3+\dots \dots .+{}^{10}{C}_{10}\cdot 2^{10}=(1+2{)}^{10}=3^{10}$