In the expansion of (1+x)^p(1-x)^q , where p and q are positive integers with p < q , the sum of the indices p+q is 7 and the coefficient of x^2 is 1 . The coefficient of x^3 is :

Step-by-step solution

The expansion of (1+x)^p(1-x)^q can be written as: (1 + px + p(p-1) 2 x^2 + ) (1 - qx + q(q-1) 2 x^2 - ) The coefficient of x^2 is obtained by multiplying the relevant terms: Coefficient of x^2 = p(p-1) 2 - pq + q(q-1) 2 This can be simplified as: p^2 - p - 2pq + q^2 - q 2 = (p-q)^2 - (p+q) 2 We are given that p+q = 7 and the coefficient of x^2 is 1 . Substituting these values: (p-q)^2 - 7 2 = 1 (p-q)^2 - 7 = 2 (p-q)^2 = 9 Since p Solving the system of equations: p + q = 7 -p + q = 3 Adding both equations gives 2q