$\begin{pmatrix} 30 \ 0 \end{pmatrix}\begin{pmatrix} 30 \ 10 \end{pmatrix} - \begin{pmatrix} 30 \ 1 \end{pmatrix}\begin{pmatrix} 30 \ 11 \end{pmatrix} + ...\begin{pmatrix} 30 \ 20 \end{pmatrix}\begin{pmatrix} 30 \ 30 \end{pmatrix}is\ equal\ to$

Question

Dr. Abubacker · Accepted Answer

(c)
Given, $A = \ ^{30}C_{0} \bullet \ ^{30}C_{10} - \ ^{30}C_{1} \bullet \ ^{30}C_{11} + \ ^{30}C_{2} \bullet \ ^{30}C_{12} + \ldots + \ ^{30}C_{20} \bullet \ ^{30}C_{30}$
$=$ coefficient of $x^{20}\ in\$ ${(1 + x)}^{30}{(1 - x)}^{30}$
$=$ coefficient of $x^{20}\ in\$ $\left( 1 + x^{2} \right)^{30}$
$\ \ \ \ \ \ \ \ \ \ \ \  =$ coefficient of $x^{20}\ in\ \sum_{r = 0}^{30}{\ \ }( - 1)^{r}\ \ ^{30}C_{r}(x^{2})^{r}$
$\ \ \ \ \ \ \ \ \ \ \ \  = {( - 1)}^{10} \bullet \ ^{30}C_{10}${for coefficient of $x^{20}$, let $r = 10$}
$\ \ \ \ \ \ \ \ \ \ \ \  = \ ^{30}C_{10}$

Q. $\begin{pmatrix} 30 \\ 0 \end{pmatrix}\begin{pmatrix} 30 \\ 10 \end{pmatrix} - \begin{pmatrix} 30 \\ 1 \end{pmatrix}\begin{pmatrix} 30 \\ 11 \end{pmatrix} + ...\begin{pmatrix} 30 \\ 20 \end{pmatrix}\begin{pmatrix} 30 \\ 30 \end{pmatrix}is\ equal\ to$

Explanation

Reviewed by Dr. Abubacker

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