Question

$\binom{30}{0} \binom{30}{10}-\binom{30}{1}\binom{30}{11}+\binom{30}{2}\binom{30}{12}+...+\binom{30}{20}\binom{30}{30}$ is equal to

• A. $^{30}C_{11}$
• B. $^{60}C_{10}$
• C. $^{30}C_{10}$
• D. $^{65}C_{55}$

Answer 1

Answer: (C)

Let $A=\begin{pmatrix}30 \\ 0\end{pmatrix}\begin{pmatrix}{c}30 \\ 10\end{pmatrix}-\begin{pmatrix}30 \\ 1\end{pmatrix}\begin{pmatrix}30 \\ 11\end{pmatrix}+\begin{pmatrix}30 \\ 2\end{pmatrix}\begin{pmatrix}30 \\ 12\end{pmatrix}-\ldots+\begin{pmatrix}30 \\ 20\end{pmatrix}\begin{pmatrix}30 \\ 30\end{pmatrix}$
$\therefore A={ }^{30} C_{0} \cdot{ }^{30} C_{10}-{ }^{30} C_{1} \cdot{ }^{30} C_{11}+{ }^{30} C_{2} \cdot{ }^{30} C_{12}$
$-\ldots+{ }^{30} C_{30}{ }^{30} C_{30}$
$=$ Coefficient of $x^{20}$ in $(1+x)^{30}(1-x)^{30}$
$=$ Coefficient of $x^{20}$ in $\left(1-x^{2}\right)^{30}$
$=$ Coefficient of $x^{20}$ in $\sum_{r=0}^{30}(-1)^{r^{30}} C_{r}\left(x^{2}\right)^{r}$
$=(-1)^{10}{ }^{30} C_{10} $ [for coefficient of $x^{20}$, put $r=10$ ]
$={ }^{30} C_{10}$