The constant term in the expansion of ( 2 x + 1 x 7 + 3 x 2 ) 5 is _______. [2023]

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Solution

Q.
The constant term in the expansion of ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ is _______. [2023]

Ans.
(1080)

$General term = \frac{5!}{r_{1}! r_{2}! r_{3}!} {(2 x)}^{r_{1}} {(x^{- 7})}^{r_{2}} {(3 x^{2})}^{r_{3}}$

$= \frac{5!}{r_{1}! r_{2}! r_{3}!} 2^{r_{1}} \cdot 3^{r_{3}} \cdot x^{r_{1} - 7 r_{2} + 2 r_{3}}$

For the constant term, $r_{1} - 7 r_{2} + 2 r_{3} = 0$ and $r_{1} + r_{2} + r_{3} = 5$

By hit and trial, $r_{1} = 1$ , $r_{2} = 1$ and $r_{3} = 3$ .

$∴$ $Constant term = \frac{5!}{1! 1! 3!} 2^{1} \cdot 3^{3}$

$= \frac{120}{6} \times 2 \times 27 = 20 \times 2 \times 27 = 1080$

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