If ${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7$ $+\mathrm{......}+10{\left(11\right)}^9=k{\left(10\right)}^9$ then $k=$  

${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7+\mathrm{.....}+10{\left(11\right)}^9=k{\left(10\right)}^9$ then k =

$1.s={1.10}^9+{\mathrm{2.11.10}}^8+{3.11}^2{.10}^7+\mathrm{.......}+{\mathrm{9.10.11}}^8+{10.11}^9$

$\frac{11}{10}s={11.10}^8+{2.11}^2{10}^7+\mathrm{.....}+{9.11}^9+\mathrm{...}+{11}^{10}$

$\left(1-\frac{11}{10}\right)s={10}^9+{11.10}^8+{11}^2{.10}^7+\mathrm{....}+{11}^9-{11}^{10}$

$\frac{-1}{10}s={10}^9\frac{\left(\left({\frac{11}{10}}^{10}\right)-1\right)}{\frac{11}{10}-1}-{11}^{10}$

$\frac{-1}{10}s={10}^{10}\left(\frac{{11}^{10}}{{10}^{10}}-1\right)-{11}^{10}=\cancel{{11}^{10}}-{10}^{10}-\cancel{{11}^{10}}$

$+s=+{10}^{10}\times 10={10}^{11}=k{.10}^9$

$\cancel{{10}^9}{.10}^2=k.\cancel{{10}^9}$

$k=100$