${ }^{11}{C}_{10}{\cdot }^9{C}_1{+}^{11}{C}_9{\cdot }^9{C}_2+\dots \dots .{+}^{11}{C}_2{\cdot }^9{C}_9=$=

$(1+x{)}^{11}{=}^{11}{C}_0{+}^{11}{C}_{1}x{+}^{11}{C}_2{x}^2$$+\dots \dots \dots {+}^{11}{C}_{11}{x}^{11}-\left(\mathrm{I}\right)$

$(1+x{)}^9{=}^9{C}_0{+}^9{C}_{1}x{+}^9{C}_2{x}^2+\dots \dots \dots {+}^9{C}_9{x}^9\text{-(II)}$

$\text{On multiply (I) \& (II) and compare coefficient}$  $\text{of }{x}^{11}\text{on both sides and put }x=1$
${ }^{20}{C}_{11}{=}^{11}{C_{11}}^9{C}_0{+}^{11}{C_{10}}^9{C}_1+\dots \dots ..{+}^{11}{C}_2^9{C}_9$

$\therefore { }^{20}{C}_9-1{=}^{11}{C_{10}}^9{C}_1+\dots \dots .{+}^{11}{C_2}^9{C}_9$