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Question

Mathematics Question on Binomial theorem

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

Coefficient of x in (1 + x)p (1 – x)q
−pC0 qC1+pC1 qC0=−3^{−p}C_0\ ^{q}C_1+ ^{p}C_1\ ^qC_0=−3
⇒p−q=−3⇒ p−q=−3
Coefficient of x2 in (1 + x)p (1 – x)q
pC0 qC2−pC1 qC1\+pC2 qC0=−5^pC_0\ ^qC_2− ^pC_1\ ^qC_1 \+ ^pC_2\ ^qC_0=−5
q(q−1)2−pq+p(p−1)2=−5\frac{q(q−1)}{2}−pq+\frac{p(p−1)}{2}=−5
q2−q2−(q−3)q+(q−3)(q−4)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
=−11C3+8C111C2−8C211C1+8C3=^{−11}C_3+ ^8C_1 ^{11}C_2− ^8C_2 ^{11}C_1+ ^8C_3
=23
So, the answer is 23.