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 _____________.

Solution

Maths sir · 5 steps

Turn the two given coefficients into two equations in p and q, solve, then build the x³ coefficient.

1. Coefficient of x. Expand (1+x)ᵖ(1-x)^q = (Σ C(p, r)xʳ)(Σ C(q, s)(-x)ˢ). The x term comes from px and -qx, so p - q = -3.
2. Coefficient of x squared. The x² terms give C(p, 2) - pq + C(q, 2). Combine these over a common denominator of 2.

   C(p, 2) - pq + C(q, 2) = ((p-q)² - (p+q))/2 = (9 - (p+q))/2 = -5

3. Find p and q. This gives p + q = 19. With p - q = -3, you get p = 8 and q = 11, both within 15.
4. Coefficient of x cubed. It is C(p, 3) - C(p, 2)q + pC(q, 2) - C(q, 3) = 56 - 28 × 11 + 8 × 55 - 165, which is 56 - 308 + 440 - 165.
5. So. The coefficient of x³ is 23.

The common slip: 353. Forgetting that (-x)³ carries a minus sign makes the last term +165, which gives 353.

The idea to keep. In (1+x)ᵖ(1-x)^q, pair the terms whose powers add to the one you need, and give every odd power of (-x) a minus sign. Writing p-q and p+q often turns the equations linear.

Answer from the NTA final answer key. Solution by PrepWiser.