The coefficient of x^{256} in the expansion o | vedclass

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(C) Given expression: $y = (1-x)(1-x)^{100}(x^{2}+x+1)^{100}$Since $(1-x)(1+x+x^{2}) = (1-x^{3})$,we can rewrite the expression as:$y = (1-x)((1-x)(1+

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$^{100}C_{15}$

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(C) Given expression: $y = (1-x)(1-x)^{100}(x^{2}+x+1)^{100}$Since $(1-x)(1+x+x^{2}) = (1-x^{3})$,we can rewrite the expression as:$y = (1-x)((1-x)(1+x+x^{2}))^{100} = (1-x)(1-x^{3})^{100}$Expanding this:$y = (1-x^{3})^{100} - x(1-x^{3})^{100}$We need the coefficient of $x^{256}$.In $(1-x^{3})^{100}$,the general term is $^{100}C_{r}(-1)^{r}(x^{3})^{r} = ^{100}C_{r}(-1)^{r}x^{3r}$.For $x^{256}$,$3r$ cannot be $256$ (as $256$ is not divisible by $3$).In $-x(1-x^{3})^{100}$,we need the coefficient of $x^{255}$ in $(1-x^{3})^{100}$.Setting $3r = 255$,we get $r = 85$.The term is $-1 	imes (^{100}C_{85}(-1)^{85}x^{255}) = -1 	imes (^{100}C_{85} 	imes -1)x^{255} = ^{100}C_{85}x^{255}$.Since $^{100}C_{85} = ^{100}C_{100-85} = ^{100}C_{15}$,the coefficient is $^{100}C_{15}$.

The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}(x^{2}+x+1)^{100}$ is:

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