Explanation

(c)
Let $A = \begin{pmatrix}
30 \\
0
\end{pmatrix}\begin{pmatrix}
30 \\
10
\end{pmatrix} - \begin{pmatrix}
30 \\
1
\end{pmatrix}\begin{pmatrix}
30 \\
11
\end{pmatrix}x$
$+ \begin{pmatrix}
30 \\
2
\end{pmatrix}\begin{pmatrix}
30 \\
12
\end{pmatrix} - ... + \begin{pmatrix}
30 \\
20
\end{pmatrix}\begin{pmatrix}
30 \\
30
\end{pmatrix}$
or$A = \ ^{30}C_{0}.\ ^{30}C_{10} - \ ^{30}C_{1}.\ ^{30}C_{11}$
$+ \ ^{30}C_{2}.\ ^{30}C_{12} - ... + \ ^{30}C_{20}.\ ^{30}C_{30}$
$= coefficient\ of\ x^{20}\ in\ {(1 + x)}^{30}{(1 - x)}^{30}$
$= coefficient\ of\ x^{20}\ in\ {(1 - x^{2})}^{30}$
$= coefficient\ of\ x^{20}\ in\ \sum_{r = 0}^{30}{{( - 1)}^{r}\ ^{30}C_{r}{(x^{2})}^{r}}$
$= {( - 1)}^{10}\ ^{30}C_{10}$ (for coefficient of $x^{20}$, let $r = 10$)
$= \ ^{30}C_{10}$