In the expansion of (1 + x + x^3 + x^4)^{10}, | vedclass

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(D) Given expression: $(1 + x + x^3 + x^4)^{10} = (1 + x)(1 + x^3)^{10}$ is incorrect. Correct factorization: $(1 + x + x^3 + x^4)^{10} = ((1 + x) + x

Vedclass · Accepted Answer

$310$

Vedclass · Answer

(D) Given expression: $(1 + x + x^3 + x^4)^{10} = (1 + x)(1 + x^3)^{10}$ is incorrect. Correct factorization: $(1 + x + x^3 + x^4)^{10} = ((1 + x) + x^3(1 + x))^{10} = ((1 + x)(1 + x^3))^{10} = (1 + x)^{10}(1 + x^3)^{10}$.Expanding both parts:$(1 + x)^{10} = 1 + ^{10}C_1 x + ^{10}C_2 x^2 + ^{10}C_3 x^3 + ^{10}C_4 x^4 + \dots$$(1 + x^3)^{10} = 1 + ^{10}C_1 x^3 + ^{10}C_2 x^6 + \dots$To find the coefficient of $x^4$,we multiply terms from the two expansions:$(1 	imes 	ext{coefficient of } x^4 	ext{ in } (1+x^3)^{10}) + (^{10}C_1 x 	imes 	ext{coefficient of } x^3 	ext{ in } (1+x^3)^{10}) + (^{10}C_2 x^2 	imes 0) + (^{10}C_3 x^3 	imes 0) + (^{10}C_4 x^4 	imes 1)$.Since $(1+x^3)^{10}$ has no $x^4$ term,we only consider the $x^3$ term from the second bracket:Coefficient of $x^4 = (1 	imes 0) + (^{10}C_1 	imes ^{10}C_1) + (^{10}C_4 	imes 1) = 0 + 10 	imes 10 + 210 = 100 + 210 = 310$.

In the expansion of $(1 + x + x^3 + x^4)^{10}$,the coefficient of $x^4$ is

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