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 , th…

Question

If the Coefficient of ${x}^{30}$ in the expansion of ${(1+\frac{1}{x})}^{6}{(1+{x}^{2})}^{7}{(1-{x}^{3})}^{8};x
e 0$ is $\alpha $, then $|\alpha |$ equals \_\_\_\_\_\_\_\_\_\_\_.

Answer: 678

Accepted Answer

$\begin{gathered} {x}^{30}\frac{(1+x{)}^{6}{(1+{x}^{2})}^{7}{(1-{x}^{3})}^{8}}{{x}^{6}} \ \Rightarrow {x}^{36}(1+x{)}^{6}{(1+{x}^{2})}^{7}{(1-{x}^{3})}^{8} \ \Rightarrow ={}^{6}{C}_{{r}_{1}}{}^{7}{C}_{{r}_{2}}{}^{8}{C}_{{r}_{3}}(-1{)}^{{r}_{3}}{x}^{{r}_{1}+2{r}_{2}+3{r}_{3}} \ \Rightarrow {r}_{1}+2{r}_{2}+3{r}_{3}=36 \end{gathered}$

$\begin{gathered} {r}_{1} \quad  {r}_{2} \quad  {r}_{3} \ 0 \quad  6 \quad  8 \ 2 \quad  5 \quad  8 \ 4 \quad  4 \quad  8 \ 6 \quad  3 \quad  8 \end{gathered}$

${r}_{1}+2{r}_{2}=12\, ($ Taking ${r}_{3}=8)$

$\begin{gathered} \begin{gathered} {r}_{1} \quad {r}_{2} \quad {r}_{3} \ 1 \quad 7 \quad 7 \ 3 \quad 6 \quad 7 \ 5 \quad 5 \quad 7 \end{gathered}$

${r}_{1}+2{r}_{2}=15($ Taking ${r}_{3}=7)$

$\begin{gathered} {r}_{1} \quad {r}_{2} \quad {r}_{3} \ 4 \quad 7 \quad 6 \ 6 \quad 6 \quad 6 \end{gathered} \end{gathered}$

$\begin{gathered} =7+(15	imes 21)+(15	imes 35)+(35) \ -(6	imes 8)-(20	imes 7	imes 8)-(6	imes 21	imes 8)+(15	imes 28) \ +(7	imes 28)=-678=\alpha \ |\alpha |=678 \end{gathered}$

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