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$\left(1-x-x^{2}+x^{3}\right)^{6}$ के प्रसार में $x^{7}$ का गुणांक है:
$-132$
$-144$
$132$
$144$
Solution
$\left(1-x-x^{2}+x^{3}\right)^{6}=\left((1-x)\left(1-x^{2}\right)\right)^{6}$
$=(1-x)^{12} \cdot(1+x)^{6}$
Coefficient of $x^{n}$ in $(1+x)^{6}=^{6} C_{n}$
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 $(1+x)^{6}+$
$-\left(^{12} C_{1}^{6} C_{6}\right)+\left(^{12} C_{2}^{6} C_{5}\right)-\left(^{12} C_{3}^{6} C_{4}\right)+\left(^{12} C_{4}^{6} C_{3}\right)$
$-\left(^{12} C_{5} \cdot^{6} C_{2}\right)+\left(^{12} C_{6} \cdot^{6} C_{1}\right)-\left(^{12} C_{7} \cdot^{6} C_{0}\right)$
$=-12+(66 \times 6)-(220 \times 15)+(495 \times 20)-(792 \times 15)+(132 \times 42)-792$
$=-144$
