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

 

We have $\left.\sum _{0\le } \sum _{i<j\le 10}{ }^{10}{C_j}^j{C}_i\right)$

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