If the Coefficient of x^30 in the expansion of (1+(1 / x))^6(1+x^2)^7(1-x^3)^8 ; x ≠ 0 is alpha, then |alpha| equals ___________.

Coefficient of x^30 in ((1+x)^6(1+x^2)^7(1-x^3)^8 / x^6) \ => Coefficient of x^36 in (1+x)^6(1+x^2)^7(1-x^3)^8 \ => General term = ^6 C_r_1 ^7 C_r_2 ^8 C_r_3(-1)^r_3 x^r_1+2 r_2+3 r_3 \ => r_1+2 r_2+3 r_3=36\ Case-I : \|c|c|c| hline r_1 r_2 r_3 \ hline 0 6 8 \ hline 2 5 8 \ hline 4 4 8 \ hline 6 3 8 \ hline r_1+2 r_2=12 (. Taking .r_3=8) Case-II : \ \|l|l|l| hline r_1 r_2 r_3 \ hline 1 7 7 \ hline 3 6 7 \ hline 5 5 7 \ hline r_1+2 r_2=15(. Taking .r_3=7) Case-III : \ \|l|l|l| hline r_1 r_2 r_3 \ hline 4 7 6 \ hline 6 6 6 \ hline Coefficient=7+(15 x 21)+(15 x 35)+(35) \ -(6 x 8)-(20 x 7 x 8)-(6 x 21 x 8)+(15 x 28) \ +(7 x 28)=-678=alpha \ |alpha|=678\

If the Coefficient of x30 in the expansion of 1frac 1x6... | null (Mathematics)

\begin{aligned} & \text { Coefficient of } x^{30} \text { in } \frac{(1+x)^6\left(1+x^2\right)^7\left(1-x^3\right)^8}{x^6} \\\\ & \Rightarrow \text { Coefficient of } x^{36} \text { in }(1+x)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 \\\\ & \Rightarrow \text { General term }={ }^6 C_{r_1}{ }^7 C_{r_2}{ }^8 C_{r_3}(-1)^{r_3} x^{r_1+2 r_2+3 r_3} \\\\ & \Rightarrow r_1+2 r_2+3 r_3=36\end{aligned}

\text { Case-I : } \begin{array}{|c|c|c|} \hline \mathrm{r}_1 & \mathrm{r}_2 & \mathrm{r}_3 \\ \hline 0 & 6 & 8 \\ \hline 2 & 5 & 8 \\ \hline 4 & 4 & 8 \\ \hline 6 & 3 & 8 \\ \hline \end{array}

r_1+2 r_2=12 \quad\left(\right.

Taking

\left.r_3=8\right)

\begin{aligned} &\text { Case-II :}\\\\ &\begin{array}{|l|l|l|} \hline r_1 & r_2 & r_3 \\ \hline 1 & 7 & 7 \\ \hline 3 & 6 & 7 \\ \hline 5 & 5 & 7 \\ \hline \end{array} \end{aligned}

r_1+2 r_2=15\left(\right.

Taking

\left.r_3=7\right)

\begin{aligned} &\text { Case-III : }\\\\ &\begin{array}{|l|l|l|} \hline r_1 & r_2 & r_3 \\ \hline 4 & 7 & 6 \\ \hline 6 & 6 & 6 \\ \hline \end{array} \end{aligned}

\begin{aligned} & \text { Coefficient}=7+(15 \times 21)+(15 \times 35)+(35) \\\\ & -(6 \times 8)-(20 \times 7 \times 8)-(6 \times 21 \times 8)+(15 \times 28) \\\\ & +(7 \times 28)=-678=\alpha \\\\ & |\alpha|=678\end{aligned}

Explanation

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