The value of  $\sum _{\text{r}=0}^6{(}^6{\text{C}}_r\cdot {}^6{\text{C}}_{\text{6}-r})$  is equal to

  $\sum _{R=0}^6{}^{6}C_r.{}^{6}C_{6-r}$    $={}^{6}C_0.{}^{6}C_6+{}^{6}C_1.{}^{6}C_5+\mathrm{.......}+{}^{6}C_6.{}^{6}C_0$
  
 Now, 

 $$\begin{gathered} {(1+x)}^6{(1+x)}^6 =({}^{6}C_0+{}^{6}C_{1}x+{}^{6}C_2{x}^2+\mathrm{....}+{}^{6}C_6{x}^6) \\ ({}^{6}C_0+{}^{6}C_{1}x+{}^{6}C_2{x}^2+\mathrm{....}+{}^{6}C_6{x}^6) \end{gathered}$$
   Comparing coefficient of   both sides
  $$\begin{gathered} {}^{6}C_0.{}^{6}C_6+{}^{6}C_1+{}^{6}C_5+\mathrm{.......}+{}^{6}C_6.{}^{6}C_0={}^{12}C_6 \\ =924 \end{gathered}$$