What is the coefficient?

Question

$(1-x)^21(1+x^2 +x)^20$
find the coefficient of $x^17$

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Answer 1 · 2018-02-20T15:59:12

$0$

The coefficient of $x^17$ is $0$ because

$(1-x)^21(1+x^2 +x)^20 = (1-x)^21((x^3-1)/(x-1))^20 = (1-x)(x^3-1)^20$

Now the powers for $x$ are $3k$ and $3k+1$ and as we can verify

$3k=17$ or $3k+1=17$ does not have integer solution for $k$