If (1 - x^3)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r , then 9a₉ a₁₀ is equal to __________.

Step-by-step solution

The given equation is: (1 - x^3)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r Using the algebraic identity 1 - x^3 = (1-x)(1+x+x^2) , the left hand side can be written as: (1-x)¹⁰(1+x+x^2)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r Dividing both sides by (1-x)³⁰ : (1-x)¹⁰(1+x+x^2)¹⁰ (1-x)³⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 2r ( 1+x+x^2 (1-x)^2 )¹⁰ = _ r=0 ¹⁰ a_r ( x (1-x)^2 )^r Let y = x (1-x)^2 . The term inside the bracket on the left hand side can be simplified as: 1+x+x^2 (1-x)^2 = 1-2x+x^2+3x (1-x)^2 = (1-x)^2+3x (1-x)^2 = 1 + 3 ( x (1-