The value of ∑ r = 0 20 C 6 50 - r is equal to:

$\sum _{r=0}^{20}{}^{50-r}C_{\mathit{6}}\mathit{=}$

${\mathit{=}}^{\mathit{50}}{C}_{\mathit{6}}{+}^{\mathit{49}}{C}_{\mathit{6}}{+}^{\mathit{48}}{C}_{\mathit{6}}+\mathit{.}\mathit{..}\mathit{.}{+}^{\mathit{30}}{C}_{\mathit{6}}$

$={-}^{30}{\mathrm{C}}_7{+}^{30}{\mathrm{C}}_7{+}^{30}{\mathrm{C}}_6{+}^{31}{\mathrm{C}}_6{+}^{32}{\mathrm{C}}_6+....{+}^{50}{\mathrm{C}}_6$

$\mathit{=}{\mathit{-}}^{\mathit{30}}{C}_{\mathit{7}}{+}^{\mathit{31}}{C}_{\mathit{7}}{+}^{\mathit{31}}{C}_{\mathit{6}}{+}^{\mathit{32}}{C}_{\mathit{6}}+\mathit{.}\mathit{...}{+}^{\mathit{50}}{C}_{\mathit{6}}$

$={-}^{30}{\mathrm{C}}_7{+}^{32}{\mathrm{C}}_7{+}^{32}{\mathrm{C}}_6+....{+}^{50}{\mathrm{C}}_6$

$={-}^{30}{C}_7{+}^{51}{C}_7$.