The coefficient of x 17 in ( 1 - x ) 13 ( 1 + x + x 2 ) 12 is

Question

The coefficient of $x^{17}$ in ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$ is

Smart Achievers · Accepted Answer

(3)

$We have,$ ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$

$= (1 - x) {[(1 - x) (1 + x + x^{2})]}^{12} = (1 - x) {(1 - x^{3})}^{12}$

In the expansion, all the powers of $x$ is either in the form of $3 k$ or $(3 k + 1)$ for some $k \in ℤ$ . But 17 has the form $(3 k + 2)$ .

$∴$ Coefficient of $x^{17}$ in the given expansion is 0.

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