The coefficient of $$x18$$ in the product $$(1+x)(1$$−$$x)101+x+x29is$$

Question

The coefficient of $$x18$$ in the product $$(1+x)(1$$−$$x)101+x+x29is$$

Infinity Learn · Accepted Answer

Given expression is $$(1+x)(1$$−$$x)101+x+x29=(1+x)(1$$−$$x)(1$$−$$x)1+x+x29=1$$−$$x21$$−$$x39Now$$, coefficient of $$x18$$ in the product $$(1+x)(1$$−$$x)101+x+x29=coefficient$$ of $$y18$$ in the product $$1$$−$$x21$$−$$x39=coefficient$$ of $$y18$$ in $$(1$$ $$-$$ $$x3$$ $$)9-$$ coefficient of $$x16in$$ $$(1$$ $$-$$ $$x3)9Since$$, $$(r$$ $$+$$ $$1)th$$ term in the expansion $$of1$$−$$x39$$ is $$9Cr$$−$$x3r=9Cr($$−$$1)rx3rNow$$, for $$x18$$, $$3r$$ $$=$$ $$18$$⇒r $$=$$ $$6and$$ for $$x18$$ , $$3r=18$$⇒$$r=6and$$ for $$x16$$,$$3r=16$$⇒$$r=163$$∉N.'. Required coefficient is  $$9C6=9$$!$$6$$!$$3$$!$$=9$$×$$8$$×$$73$$×$$2=84$$

The coefficient of x¹⁸ in the product $(1 + x) (1 - x)^{10} {(1 + x + x^{2})}^{9} is$

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The coefficient of x18 in the product (1+x)(1−x)101+x+x29is

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The coefficient of x¹⁸ in the product $(1 + x) (1 - x)^{10} {(1 + x + x^{2})}^{9} is$