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The coefficient of  x^3  in   \left(\sqrt{x^5}+\frac{3}{\sqrt{x^3}}\right)^6 is

Option: 1

0


Option: 2

120


Option: 3

420


Option: 4

540


Answers (1)

best_answer

\mathrm{T}_{\mathrm{r}+1}={ }^6 \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{5 / 2}\right)^{6-\mathrm{r}}\left(\frac{3}{\mathrm{x}^{3 / 2}}\right)^r

={ }^6 \mathrm{C}_r 3^r x^{15-\frac{5 r}{2} \cdot \frac{3 r}{2}}={ }^6 \mathrm{C}_r 3^r x^{15-4 r}

Let  T_{r+1} contains  x^3.

15 - 4 r=3 or r=3

\mathrm{T}_{\mathrm{r}+1}=\mathrm{T}_{3+1}={ }^6 \mathrm{C}_3(3)^3 \mathrm{x}^{15-4(3)}\

=20 \times 27\,\, x^3=540 x^3

Coefficient of  x^3=540.

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Nehul

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