Question

what the coefficient of
${x}^{21} \: in \: { (1 + x + {x}^{2} ....... + {x}^{9} )}^{3}$

Answer 1

Firstly
simplify the expression
it an gp
hence
it can be simplified as
$\frac{ {(1 - {x}^{10}) }^{3} }{ {(1 - x)}^{3} } \\ \\ = {( 1 - {x}^{10} ) }^{3} {( 1 - x)}^{ - 3} \\ simplifying \\ (1 + 3 {x}^{20} - 3 {x}^{10} ) {(1 - x)}^{ - 3} \\ coefficent \: of \: {x}^{21} = \\ \binom{23}{21} + 3 \times \binom{3}{1} - 3 \times \binom{13}{11} \\ = 253 + 9 - 234 \\ = 28 \\ hence \: 28 \: will \: be \: the \: coefficient \: of \: {x}^{21}$