The coefficient of x⁷ in the expression $(1+\mathrm{x}{)}^{10}+\mathrm{x}(1+\mathrm{x}{)}^9+{\mathrm{x}}^2(1+\mathrm{x}{)}^8+\dots +{\mathrm{x}}^{10}$ is

Given expression $(1+\mathrm{x}{)}^{10}+\mathrm{x}(1+\mathrm{x}{)}^9+{\mathrm{x}}^2(1+\mathrm{x}{)}^8+\dots +{\mathrm{x}}^{10}$

The above expression is geometric series having first term,$\mathrm{a}=(1+\mathrm{x}{)}^{10}$

Common ratio,$\mathrm{r}=\frac{\mathrm{x}}{1+\mathrm{x}}$ and number of terms, n = 11

sum of geometric series $=\frac{\mathrm{a}\left(1-{\mathrm{r}}^{\mathrm{n}}\right)}{1-\mathrm{r}}=\frac{(1+\mathrm{x}{)}^{10}\left(1-{\left(\frac{\mathrm{x}}{1+\mathrm{x}}\right)}^{11}\right)}{1-\frac{\mathrm{x}}{1+\mathrm{x}}}$

                                         $\begin{array}{c}=\frac{(1+\mathrm{x}{)}^{10}\left(\frac{(1+\mathrm{x}{)}^{11}-{\mathrm{x}}^{11}}{(1+\mathrm{x}{)}^{11}}\right)}{\frac{1+\mathrm{x}-\mathrm{x}}{1+\mathrm{x}}}\\ =(1+\mathrm{x}{)}^{11}-{\mathrm{x}}^{11}\end{array}$

The coefficient of x7 an the given expression
= coefficient of x7 in the expansion of $\left[(1+\mathrm{x}{)}^{11}-{\mathrm{x}}^{11}\right]$
= coefficient of x7 in the expansion of $(1+\mathrm{x}{)}^{11}$

$\begin{array}{c}{=}^{11}{\mathrm{C}}_7=\frac{11\times 10\times 9\times 8}{4\times 3\times 2}\\ =11\times 10\times 3=330\end{array}$