Find the coefficient of the term which contains the 10th power of x in the expansion of ${⎛ ⎜ ⎝ x^{\frac{1}{3}} + \frac{1}{x^{3} \sqrt{x}} ⎞ ⎟ ⎠}^{40}$ .

870

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680

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780

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none of these

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Solution

The correct option is D none of these
Given, ${(x^{\frac{1}{3}} + \frac{1}{x^{3} \sqrt{x}})}^{40}$

General term, $T_{r + 1} =^{n} C_{r} x^{n - r} y^{r}$

$T_{r + 1} =^{40} C_{r} (x^{\frac{1}{3}})^{40 - r} {(\frac{1}{x^{3} \sqrt{x}})}^{r}$

$T_{r + 1} =^{40} C_{r} (x)^{\frac{40}{3} - \frac{r}{3}} {⎛ ⎜ ⎝ \frac{1}{x^{\frac{7}{2}}} ⎞ ⎟ ⎠}^{r}$

$T_{r + 1} =^{40} C_{r} (x)^{\frac{40}{3} - \frac{23 r}{6}}$

We need to find the term which contains $10^{t h}$ power of x
So, $\frac{40}{3} - \frac{23 r}{6} = 10$

$\frac{80 - 23 r}{6} = 10$
$80 - 23 r = 60$
$20 = 23 r$
$r = \frac{20}{23}$
Which is not possible.

Hence, correct option is D.

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