જો left(1+frac{1}{x}right)^6left(1+x^2r | vedclass

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Question

coeff of $x^{30}$ in $\frac{(x+1)^6\left(1+x^2\right)^7\left(1-x^3\right)^8}{x^6}$

coeff. of  $x^{36}$ in $(1+x)^6\left(1+x^2\right)^7\left(1-

Vedclass · Accepted Answer

$678$

Vedclass · Answer

coeff of $x^{30}$ in $\frac{(x+1)^6\left(1+x^2ight)^7\left(1-x^3ight)^8}{x^6}$

coeff. of  $x^{36}$ in $(1+x)^6\left(1+x^2ight)^7\left(1-x^3ight)^8$

General term

${ }^6 \mathrm{C}_{\mathrm{r}_1}{ }^7 \mathrm{C}_{\mathrm{r}_2}{ }^8 \mathrm{C}_{\mathrm{r}_3}(-1)^{\mathrm{I}_3} \mathrm{x}^{\mathrm{I}_1+2 \mathrm{r}_2+3 \mathrm{r}_3}$

$\mathrm{r}_1+2 \mathrm{r}_2+3 \mathrm{r}_3=36$

Case-$I$ :    $r_1+2 r_2=12\left(ight.$ Taking $\left.r_3=8ight)$


	
		
			$r_1$
			$r_2$
			$r_3$
		
		
			$0$
			$6$
			$8$
		
		
			$2$
			$5$
			$8$
		
		
			$4$
			$4$
			$8$
		
		
			$6$
			$3$
			$8$
		
	


case-$II$       $r_1+2 r_2=15\left(ight.$ Taking $\left.r_3=7ight)$


	
		
			$r_1$
			$r_2$
			$r_3$
		
		
			$1$
			$7$
			$7$
		
		
			$3$
			$6$
			$7$
		
		
			$5$
			$5$
			$7$
		
	


case-$III$      $r_1+2 r_2=18\left(ight.$ Taking $\left.r_3=6ight)$


	
		
			$r_1$
			$r_2$
			$r_3$
		
		
			$4$
			$7$
			$6$
		
		
			$6$
			$6$
			$6$
		
	


$\begin{aligned} & 	ext { Coeff. }=7+(15 	imes 21)+(15 	imes 35)+(35) \ & -(6 	imes 8)-(20 	imes 7 	imes 8)-(6 	imes 21 	imes 8)+(15 	imes 28) \ & +(7 	imes 28)=-678=\alpha \ & |\alpha|=678\end{aligned}$

જો $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ ના વિસ્તરણમાં $x^{30}$ નો સહગુણક $\alpha$ હોય, તો $|\alpha|=$......................

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