The coefficient of x^{50} in the expansion of | vedclass

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(A) Given expression: $(1+x)^{101}(1-x+x^2)^{100}$ $= (1+x) \cdot (1+x)^{100} \cdot (1-x+x^2)^{100}$ $= (1+x) \cdot [(1+x)(1-x+x^2)]^{100}$ $= (1+x)(1

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(A) Given expression: $(1+x)^{101}(1-x+x^2)^{100}$ $= (1+x) \cdot (1+x)^{100} \cdot (1-x+x^2)^{100}$ $= (1+x) \cdot [(1+x)(1-x+x^2)]^{100}$ $= (1+x)(1+x^3)^{100}$ $= (1+x^3)^{100} + x(1+x^3)^{100}$ We need the coefficient of $x^{50}$. In the expansion of $(1+x^3)^{100}$,the powers of $x$ are of the form $3k$,where $k$ is an integer. Since $50$ is not a multiple of $3$,the coefficient of $x^{50}$ in $(1+x^3)^{100}$ is $0$. Similarly,for $x(1+x^3)^{100}$,we need the coefficient of $x^{49}$ in $(1+x^3)^{100}$. Since $49$ is not a multiple of $3$,the coefficient of $x^{49}$ in $(1+x^3)^{100}$ is $0$. Therefore,the coefficient of $x^{50}$ is $0 + 0 = 0$.

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

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