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Binomial Coefficients

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Question

Find the term which has the exponent of x as 8 in the expansion of ${⎛ ⎜ ⎝ x^{\frac{5}{2}} - \frac{3}{x^{3} \sqrt{x}} ⎞ ⎟ ⎠}^{10}$

A

T_{2}

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B

T_{3}

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C

T_{4}

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D

Does not exist

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Solution

The correct option is C Does not exist
Given, ${⎛ ⎜ ⎝ x^{\frac{5}{2}} - \frac{3}{x^{3} \sqrt{x}} ⎞ ⎟ ⎠}^{10}$

General term,

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

$T_{r + 1} =^{10} C_{r} (x^{\frac{5}{2}})^{10 - r} {(- \frac{3}{x^{3} \sqrt{x}})}^{r}$

$T_{r + 1} =^{10} C_{r} (x)^{25 - \frac{5 r}{2}} \frac{(- 3)^{r}}{x^{\frac{7}{2} r}}$

$T_{r + 1} =^{10} C_{r} (- 3)^{r} (x)^{25 - 6 r}$

given the exponent of x as $8$

$25 - 6 r = 8$

$25 - 8 = 6 r$

$6 r = 17$

$r = \frac{17}{6}$

since r is a fraction number which is not possible. Hence, the term which has the exponents of x as $8$ in the expansion of ${⎛ ⎜ ⎝ x^{\frac{5}{2}} - \frac{3}{x^{3} \sqrt{x}} ⎞ ⎟ ⎠}^{10}$
does not exist.

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