JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Binomial Theorem

If the coefficients of x and

x^{2}

in the expansion of

(1 + x)^{p} (1 - x)^{q}, p, q \leq 15

, are -3 and -5 respectively, then the coefficient of

x^{3}

is equal to ______.

Answer: 23

Solution

Key concept: Binomial expansion and coefficient comparison.

For

(1 + x)^{p} (1 - x)^{q}

, expand both binomials and collect coefficients.

Coefficient of x:

p - q = - 3

... (1)

**Coefficient of

x^{2}

:**

- pq + \frac{p ( p - 1 )}{2} + \frac{q ( q - 1 )}{2} = - 5

... (2)

Solving (1) and (2):

p = 8, q = 11

**Coefficient of

x^{3}

:**

-^{11} C_{3} +^{8} C_{3} +^{8} C_{1}^{11} C_{2} -^{8} C_{2}^{11} C_{1}

= - 165 + 56 + 8 (55) - 28 (11)

= - 165 + 56 + 440 - 308 = 23

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Integer answer