If the coefficients of x and x ^ 2 in the expansion of (1 + x) ^ p (1 - x) ^ q , p, q ≤ 15 , are - 3 and - 5 respectively, then the coefficient of x ^ 3 is equal to ______.

Correct answer: 0 Given coefficient of x in expansion of (1 + x) ^ p (1 - x) ^ q is - 3 ⇒ C0 q C1 p - C0 p C1 q = - 3 ⇒ p - q = - 3 ⋯ (1) Now coefficients of x ^ 2 in expansion of (1 + x) ^ p (1 - x) ^ q - C1q P C _ 1 + C0 q C2 P + C0 P C2 q = - 5 ⇒ - pq + (p(p - 1))/(2) + (q(q - 1))/(2) = - 5 ⋯ (2) Solving (1) and (2) p = 8, q = 11 Coefficient of x ^ 3 in expansion of (1 + x) ^ p (1 - x) ^ q will be, - C3 q + C3 p + C p _ 1 ^ q C _ 2 - C p _ 2 ^ q C _ 1 = - C3 11 + C3 8 + C1 8 C2 11 - C2 8 C1 11 = 23

JEE Main 202226 JulyMorning Shift

Exam: JEE Main
Year: 2022
Date: 26 July
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2022 · 26 July · Morning Shift · Mathematics · Binomial Theorem

If the coefficients of x and x ^ 2 in the expansion of (1 + x) ^ p...

If the coefficients of

x

and

{x}^{2}

in the expansion of

{\left(1 + x\right)}^{p}{\left(1 - x\right)}^{q}, p, q \le 15

, are

- 3

and

- 5

respectively, then the coefficient of

{x}^{3}

is equal to ______.

Correct answer

Step-by-step solution

Given coefficient of $x$ in expansion of ${\left(1 + x\right)}^{p}{\left(1 - x\right)}^{q} is$ - 3 $$ ⇒ C0 q C1 p - C0 p C1 q = - 3 $$ ⇒ p - q = - 3 ⋯ \left(1\right) $Now coefficients of$ {x}^{2} in expansion of ${\left(1 + x\right)}^{p}{\left(1 - x\right)}$ ^{q} $$ - C1q P {C}_{1} + C0 q C2 P + C0 P C2 q = - 5 $$ ⇒ - pq + \dfrac{p\left(p - 1\right)}{2} + \dfrac{q\left(q - 1\right)}{2} = - 5 ⋯ \left(2\right) $Solving \left(1\right)$ and \left(2\right) $$ p = 8, q = 11 $Coefficient of$ {x}^{3} in expansion of ${\left(1 + x\right)}^{p}{\left(1 - x\right)}^{q}$ will be,
$- C3 q + C3 p + {C p}_{1}^{q}{C}_{2} - {C p}_{2}^{q}{C}_{1}$
$= - C3 11 + C3 8 + C1 8 C2 11 - C2 8 C1 11$
$= 23$

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