Find the coefficient of (x^{3}) in the binomial expansion of ((x – frac{3}{x^{2}})^{9}).

Find the coefficient of (x^{3}) in the binomial expansion of ((x – frac{3}{x^{2}})^{9}).

Correct Answer: Option A

Explanation

(x – frac{3}{x^{2}} = x – 3x^{-2})

Let the power on x be t, so that the power on (x^{-2}) = 9 – t

((x)^{t}(x^{-2})^{9 – t} = x^{3}  implies t – 18 + 2t = 3)

(3t = 3 + 18 = 21 therefore t = 7)

To obtain the coefficient of (x^{3}), we have

(^{9}C_{7}(x)^{7}(3x^{-2))^{2} = frac{9!}{(9 – 7)! 7!}(x)^{7}(9x^{-4}))

= (frac{9 times 8 times 7!}{7! 2!} times 9(x^{3}) = 324x^{3})