QJEEM24Mafca1: JEE Main 2024 Mathematics - Class 11 JEEM24Mafca1 Solution: If the Coefficient of in the expansion of is , then equals…

Question

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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

|\alpha|

equals ___________.

Accepted Answer

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

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

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

Taking

\left.r_3=8\right)

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

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

Taking

\left.r_3=7\right)

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

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

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