The coefficient of x^{4} in the expansion of | vedclass

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(A) The expression is $(1+x+x^{2})^{10} = (1 + x(1+x))^{10}$.Using the binomial expansion $(1+y)^{n} = \sum_{k=0}^{n} {}^{n}C_{k} y^{k}$,we get:$(1+x+

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$615$

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(A) The expression is $(1+x+x^{2})^{10} = (1 + x(1+x))^{10}$.Using the binomial expansion $(1+y)^{n} = \sum_{k=0}^{n} {}^{n}C_{k} y^{k}$,we get:$(1+x+x^{2})^{10} = \sum_{k=0}^{10} {}^{10}C_{k} (x(1+x))^{k} = \sum_{k=0}^{10} {}^{10}C_{k} x^{k} (1+x)^{k}$.To find the coefficient of $x^{4}$,we consider terms where $x^{k} (1+x)^{k}$ contributes to $x^{4}$:For $k=2$: ${}^{10}C_{2} x^{2} (1+x)^{2} = {}^{10}C_{2} x^{2} (1 + 2x + x^{2}) = {}^{10}C_{2} x^{2} + 2({}^{10}C_{2}) x^{3} + {}^{10}C_{2} x^{4}$. Coefficient is ${}^{10}C_{2} = 45$.For $k=3$: ${}^{10}C_{3} x^{3} (1+x)^{3} = {}^{10}C_{3} x^{3} (1 + 3x + \dots) = {}^{10}C_{3} x^{3} + 3({}^{10}C_{3}) x^{4} + \dots$. Coefficient is $3 	imes {}^{10}C_{3} = 3 	imes 120 = 360$.For $k=4$: ${}^{10}C_{4} x^{4} (1+x)^{4} = {}^{10}C_{4} x^{4} (1 + \dots) = {}^{10}C_{4} x^{4} + \dots$. Coefficient is ${}^{10}C_{4} = 210$.Total coefficient of $x^{4} = 45 + 360 + 210 = 615$.

The coefficient of $x^{4}$ in the expansion of $(1+x+x^{2})^{10}$ is

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