Question

Find the sum of the coefficients of even powers of $x$ in the expansion of ${\left(1+x+x^2+x^3\right)}^5$.

Answer 1

Answer: 512

${\left(1+x+x^2+x^3\right)}^5={\left[(1+x)\left(1+x^2\right)\right]}^5$
$=(1+x{)}^5{\left(1+x^2\right)}^5$
$=\left({}^5{C}_0+{}^5{C}_{1}x+{}^5{C}_2{x}^2+{}^5{C}_3{x}^3+{}^5{C}_4{x}^4+{}^5{C}_5{x}^5\right)$
$\times ({}^5{C}_0+{}^5{C}_1{x}^2+{}^5{C}_2{\left(x^2\right)}^2+{}^5{C}_3{\left(x^2\right)}^3+{}^5{C}_4{\left(x^2\right)}^4+{}^5{C}_5{\left(x^2\right)}^5)$
$=\left(1+5x+10x^2+10x^3+5x^4+x^5\right)$
$\times \left(1+5x^2+10x^4+10x^6+5x^8+x^{10}\right)$
Sum of coefficients $=2^5(1+5+10)$
$=32(16)$
$=512$