The coefficient of x^{50} in (1+x)^{101} (1-x | vedclass

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Question

(C) The given expression is $(1+x)^{101} (1-x+x^2)^{100}$.We can rewrite this as $(1+x) [(1+x)(1-x+x^2)]^{100}$.Using the identity $(1+x)(1-x+x^2) = 1

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$0$

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(C) The given expression is $(1+x)^{101} (1-x+x^2)^{100}$.We can rewrite this as $(1+x) [(1+x)(1-x+x^2)]^{100}$.Using the identity $(1+x)(1-x+x^2) = 1+x^3$,the expression becomes $(1+x)(1+x^3)^{100}$.Expanding this,we get $(1+x^3)^{100} + x(1+x^3)^{100}$.In the expansion of $(1+x^3)^{100}$,the powers of $x$ are of the form $3k$,where $k$ is an integer.For the first part $(1+x^3)^{100}$,we need the coefficient of $x^{50}$. Since $50$ is not a multiple of $3$,the coefficient is $0$.For the second part $x(1+x^3)^{100}$,we need the coefficient of $x^{49}$ in $(1+x^3)^{100}$. Since $49$ is not a multiple of $3$,the coefficient is $0$.Therefore,the total coefficient of $x^{50}$ is $0 + 0 = 0$.Hence,option $(C)$ is correct.

The coefficient of $x^{50}$ in $(1+x)^{101} (1-x+x^2)^{100}$ is......

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