If the coefficients of x and x2 in the expansion of (1 + x)p (1 – x)q, p, q ≤ 15, are -3 and -5 respectively, then the coefficient of x3 is equal to…………………

Coefficient of x in ( 1 + x ) p ( 1 − x ) q = − p C 0 q C 1 + p C 1 q C 0 = − 3 ⇒ p − q = 3 Coefficient of x2 in ( 1 + x ) p ( 1 − x ) q = p C 0 q C 2 − p C 1 q C 1 − p C 2 q C 0 = − 5 ⇒ q ( q − 1 ) 2 − p q + p ( p − 1 ) 2 = − 5 ⇒ q ( q − 1 ) 2 − ( q − 3 ) q + ( q − 3 ) ( q − 4 ) 2 = − 5 ⇒ q = 1 1 , p = 8 Coefficient of x3 in ( 1 + x ) 8 ( 1 − x ) 1 1 = − 1 1 C 3 + 8 C 1 1 1 C 2 − 8 C 2 1 1 C 1 + 8 C 3 = 2 3

If the coefficients of x and x² in the expansion of (1 + x)^p (1 – x)^q, p, q $\leq$ 15, are -3 and -5 respectively, then the coefficient of x³ is equal to…………………

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Raj Pandey | Contributor-Level 9

4 months ago

Coefficient of x in ${(1 + x)}^{p} {(1 - x)}^{q} = -^{p} {C_{0}}^{q} C_{1} +^{p} {C_{1}}^{q} C_{0} = - 3 \Rightarrow p - q = 3$

Coefficient of x² in ${(1 + x)}^{p} {(1 - x)}^{q} =^{p} {C_{0}}^{q} C_{2} -^{p} {C_{1}}^{q} C_{1} -^{p} {C_{2}}^{q} C_{0} = - 5$

$\Rightarrow \frac{q (q - 1)}{2} - p q + \frac{p (p - 1)}{2} = - 5 \Rightarrow \frac{q (q - 1)}{2} - (q - 3) q + \frac{(q - 3) (q - 4)}{2} = - 5 \Rightarrow q = 11, p = 8$

Coefficient of x³ in ${(1 + x)}^{8} {(1 - x)}^{11} = -^{11} C_{3} +^{8} {C_{1}}^{11} C_{2} -^{8} {C_{2}}^{11} C_{1} +^{8} C_{3} = 23$

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