Explanation

(c)
$(1 + x)^{21} + (1 + x)^{22} + ... + (1 + x)^{30}$
$= (1 + x)^{21}\left\lbrack 1 + (1 + x)^{1} + ... + (1 + x)^{9} \right\rbrack$
$= (1 + x)^{21}\left\lbrack \frac{(1 + x)^{10} - 1}{(1 + x) - 1} \right\rbrack$
$= \frac{1}{x}\lbrack(1 + x)^{31} - (1 + x)^{21}\rbrack$
$\therefore$ Coefficient of $x^{5}$ in the given expression
$= \$Coefficient of $x^{5}$ in $\frac{1}{x}\lbrack(1 + x)^{31} - (1 + x)^{21}\rbrack$
$= \$Coefficient of $x^{6}$ in $\lbrack(1 + x)^{31} - (1 + x)^{21}\rbrack$
$= \ ^{31}C_{6} - \ ^{21}C_{6}$