The coefficient of x⁻⁹ in the expansion of ( x^2 2 - 2 x )^9 is

Step-by-step solution

Let the coefficient of x⁻⁹ is in the (r+1) th term in the expansion of ( x^2 2 - 2 x )^9 , then aligned T_ r+1 & = ^9 C_r ( x^2 2 )^ 9-r (- 2 x )^r & = ^9 C_r x^ 18-2 r 2^ 9-r (-1)^r 2^r x^r & = ^9 C_r x^ 18-3 r 2^ 9-2 r (-1)^r aligned For coefficient of x⁻⁹ , put aligned & & x^ 18-3 r & =x⁻⁹ & & 18-3 r & =-9 & & & r aligned Coefficient of x⁻⁹= ^9 C₉ 1 ⁻⁹ (-1)^9=-2^9=-512