1e:I[80107,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1f:I[76763,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1d:T43a,{"@context":"https://schema.org","@graph":[{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://www.divinejee.com"},{"@type":"ListItem","position":4,"name":"If the Coefficient of x30 in the expansion of 1frac 1x61x271","item":"https://www.divinejee.com/question/if-the-coefficient-of-x30-in-the-expansion-of-1-frac-1x61-x271x38-x-neq-0-is-alpha-then-alpha"}]},{"@type":"QAPage","mainEntity":{"@type":"Question","name":"Question If the Coefficient of x30 in the expansion of 1frac 1x61x271 on Topic","text":"If the Coefficient of $x^{30}$ in the expansion of $\\left(1+\\frac{1}{x}\\right)^6\\left(1+x^2\\right)^7\\left(1-x^3\\right)^8 ; x \\neq 0$ is $\\alpha$, then...","answerCount":1,"upvoteCount":0,"datePublished":"2026-01-16","author":{"@type":"Organization","name":"DivineJEE"},"acceptedAnswer":{"@type":"Answer","text":"Check the detailed solution on the page.","upvoteCount":0,"url":"https://www.divinejee.com/question/if-the-coefficient-of-x30-in-the-expansion-of-1-frac-1x61-x271x38-x-neq-0-is-alpha-then-alpha"}}}]}20:T553,$\begin{aligned} & \text { Coefficient of } x^{30} \text { in } \frac{(1+x)^6\left(1+x^2\right)^7\left(1-x^3\right)^8}{x^6} \\\\ & \Rightarrow \text { Coefficient of } x^{36} \text { in }(1+x)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 \\\\ & \Rightarrow \text { General term }={ }^6 C_{r_1}{ }^7 C_{r_2}{ }^8 C_{r_3}(-1)^{r_3} x^{r_1+2 r_2+3 r_3} \\\\ & \Rightarrow r_1+2 r_2+3 r_3=36\end{aligned}$

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

$r_1+2 r_2=12 \quad\left(\right.$ 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 $\left.r_3=7\right)$

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

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