Question

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 coefficient of x³ is equal to ______.

Answer 1

Correct Answer: 23

Coefficient of x in (1 + x)^p (1 – x)^q
\(^{−p}C_0\ ^{q}C_1+ ^{p}C_1\ ^qC_0=−3\)
\(⇒ p−q=−3\)
Coefficient of x² in (1 + x)^p (1 – x)^q
\(^pC_0\ ^qC_2− ^pC_1\ ^qC_1 + ^pC_2\ ^qC_0=−5\)
\(\frac{q(q−1)}{2}−pq+\frac{p(p−1)}{2}=−5\)
\(\frac{q^2−q}{2}−(q−3)q+\frac{(q−3)(q−4)}{2}=−5\)
⇒ q = 11, p = 8
Coefficient of x³ in (1 + x)⁸ (1 – x)¹¹ is
\(=^{−11}C_3+ ^8C_1 ^{11}C_2− ^8C_2 ^{11}C_1+ ^8C_3\)
=23
So, the answer is 23.