If  ${(1+x)}^n=C_0+C_{1}x+C_2{x}^2+\mathrm{....}+C_n{x}^n$ then the value of $\sum _{r=0}^n\sum _{s=0}^n\sum _{t=0}^n{C}_r{C}_s{C}_t$  is equal to

$$\begin{gathered} \sum _{r=0}^n\sum _{s=0}^n\sum _{t=0}^n{C}_r{C}_s{C}_t=\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s\left(\sum _{t=0}^n{C}_t\right) \\ =\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s{2}^n \end{gathered}$$

$=\sum _{r=0}^n{2}^n C_r\left(\sum _{s=0}^n{C}_s\right)$

$=\sum _{r=0}^n{2}^n{C}_r{2}^n$

$=\sum _{r=0}^n{2}^{2n}{C}_r$

$=2^{2n}\left(\sum _{r=0}^n{C}_r\right)$

$=2^{2n}{2}^n$  = $2^{3n}$