In the expansion of (1+x+x^2)^p(1-x)^q , where p and q are positive integers, the coefficients of x , x^2 , and x^3 are -3 , 3 , and -15 respectively. The value of p + q is :

Step-by-step solution

The given expression is (1+x+x^2)^p(1-x)^q . This can be rewritten by grouping terms as: (1+x+x^2)^p(1-x)^p(1-x)^ q-p = ((1+x+x^2)(1-x))^p(1-x)^ q-p Using the algebraic identity 1-x^3 = (1-x)(1+x+x^2) , the expression becomes: (1-x^3)^p(1-x)^ q-p Let k = q - p . The expression is (1-x^3)^p(1-x)^k . Expanding both terms up to the power of x^3 : (1-x^3)^p = 1 - px^3 + (1-x)^k = 1 - kx + k(k-1) 2 x^2 - k(k-1)(k-2) 6 x^3 + Multiplying these expansions up to x^3 : (1 - px^3 ) (1 - kx + k(k-1) 2 x^2 - k(k-1)(k-2) 6 x^3 )