$\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{l}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{l}30\\ 11\end{array}\right)+\dots +\left(\begin{array}{l}30\\ 20\end{array}\right)\left(\begin{array}{l}30\\ 30\end{array}\right)$ is equal to

Solution:

$\mathrm{Let} \mathrm{A}=\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{l}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{l}30\\ 11\end{array}\right)+\left(\begin{array}{c}30\\ 2\end{array}\right)\left(\begin{array}{l}30\\ 12\end{array}\right)-\dots +\left(\begin{array}{l}30\\ 20\end{array}\right)\left(\begin{array}{l}30\\ 30\end{array}\right)$

$\mathrm{or} \begin{array}{l}\mathrm{A}{=}^{30}{\mathrm{C}}_0{\cdot }^{30}{\mathrm{C}}_{10}{-}^{30}{\mathrm{C}}_1{\cdot }^{30}{\mathrm{C}}_{11}{+}^{30}{\mathrm{C}}_2{\cdot }^{30}{\mathrm{C}}_{12}-\dots {+}^{30}{\mathrm{C}}_{20}{\cdot }^{30}{\mathrm{C}}_{30}\\ \end{array}$

$\begin{array}{l}=\text{ Coefficient of }{\mathrm{x}}^{20}\text{ in }(1+\mathrm{x}{)}^{30}(1-\mathrm{x}{)}^{30}\\ =\text{ Coefficient of }{\mathrm{x}}^{20}\text{ in }{\left(1-{\mathrm{x}}^2\right)}^{30}\\ =\text{ Coefficient of }{\mathrm{x}}^{20}\text{ in }\sum _{\mathrm{r}=0}^{30} {(-1{)}^{\mathrm{r}}}^{30}{\mathrm{C}}_{\mathrm{r}}{\left({\mathrm{x}}^2\right)}^{\mathrm{r}}\\ \left.={(-1{)}^{10 }}^{30}{\mathrm{C}}_{10}\text{ [for coefficient of }{\mathrm{x}}^{20},\text{ let }\mathrm{r}=10\right]\\ {=}^{30}{\mathrm{C}}_{10}\end{array}$

$\begin{array}{l}\left.=(-1{)}^{10 30}{\mathrm{C}}_{10}\text{ [for coefficient of }{\mathrm{x}}^{20},\text{ let }\mathrm{r}=10\right]\\ {=}^{30}{\mathrm{C}}_{10}\end{array}$