{left( {1 - x - {x^2} + {x^3}} right)^6} નાં | vedclass

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Question

$\left(1-x-x^{2}+x^{3}\right)^{6}=\left((1-x)\left(1-x^{2}\right)\right)^{6}$

$=(1-x)^{12} \cdot(1+x)^{6}$

Coefficient of $x^{n}$ in $(1+x)^{6}=

Vedclass · Accepted Answer

$-144$

Vedclass · Answer

$\left(1-x-x^{2}+x^{3}ight)^{6}=\left((1-x)\left(1-x^{2}ight)ight)^{6}$

$=(1-x)^{12} \cdot(1+x)^{6}$

Coefficient of $x^{n}$ in $(1+x)^{6}=^{6} C_{n}$

Coefficient of $x^{n}$ in $(1-x)^{12}=(-1)^{n} \cdot^{12} C_{n}$

Coeff. of $x^{7}$ in this expansion $=$

coeff. of $x$ in $(1-x)^{12}$. coeff of $x^{6}$ in $(1+x)^{6}+$

coeff. of $x^{2}$ in $(1-x)^{12}$. coeff of $x^{5}$ in $(1+x)^{6}+$

coeff. of $x^{3}$ in $(1-x)^{12}$. coeff of $x^{4}$ in $(1+x)^{6}+$

coeff. of $x^{4}$ in $(1-x)^{12}$. coeff of $x^{3}$ in $(1+x)^{6}+$

coeff. of $x^{5}$ in $(1-x)^{12}$. coeff of $x^{2}$ in $(1+x)^{6}+$

coeff. of $x^{6}$ in $(1-x)^{12}$ coeff of $x$ in $(1+x)^{6}+$

coeff. of $x^{7}$ in $(1-x)^{12}$. coeff of $x^{0}$ in $(1+x)^{6}+$

$-\left(^{12} C_{1}^{6} C_{6}ight)+\left(^{12} C_{2}^{6} C_{5}ight)-\left(^{12} C_{3}^{6} C_{4}ight)+\left(^{12} C_{4}^{6} C_{3}ight)$

$-\left(^{12} C_{5} \cdot^{6} C_{2}ight)+\left(^{12} C_{6} \cdot^{6} C_{1}ight)-\left(^{12} C_{7} \cdot^{6} C_{0}ight)$

$=-12+(66 	imes 6)-(220 	imes 15)+(495 	imes 20)-(792 	imes 15)+(132 	imes 42)-792$

$=-144$

${\left( {1 - x - {x^2} + {x^3}} \right)^6}$ નાં વિસ્તરણમાં $x^7$ નો સહગુણક મેળવો.

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