Find the coefficient of \(x^{3}\) in the expansion of \([\frac{1}{3}(2 + x)]^{6}\).
- \(\frac{135}{729}\)
- \(\frac{149}{729}\)
- \(\frac{152}{729}\)
-
\(\frac{160}{729}\)
The correct answer is: D
Explanation
\([\frac{1}{3}(2 + x)]^{6} = (\frac{2}{3} + \frac{x}{3})^{6}\)
The coefficient of \(x^{3}\) is
\(^{6}C_{3}(\frac{2}{3})^{3}(\frac{1}{3})^{3}x^{3} = (\frac{6!}{3!3!})(\frac{8}{27})(\frac{1}{27})x^{3}\)
= \(\frac{160}{729}\)