The coefficient of x 5 in the expansion of ( 2 x 3 - 1 3 x 2 ) 5 is [2023]

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Solution

Q.
The coefficient of $x^{5}$ in the expansion of ${(2 x^{3} - \frac{1}{3 x^{2}})}^{5}$ is [2023]

1 $\frac{26}{3}$

2 9

3 8

4 $\frac{80}{9}$

Ans.
(4)

${(2 x^{3} - \frac{1}{3 x^{2}})}^{5}$

$T_{r + 1} = {}^{5}C_{r} {(2 x^{3})}^{5 - r} {(\frac{- 1}{3 x^{2}})}^{r} = {}^{5}C_{r} \cdot 2^{5 - r} \cdot {(\frac{- 1}{3})}^{r} \cdot x^{15 - 3 r - 2 r}$

$As, 15 - 5 r = 5 \Rightarrow r = 2$

$Coefficient of x^{5} = {}^{5}C_{2} 2^{3} \times {(\frac{- 1}{3})}^{2} = \frac{80}{9}$

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