Find the 8th term in the expansion of ${(x^{3 / 2} y^{1 / 2} - x^{1 / 2} y^{3 / 2})}^{10}$ .

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Solution

We need to find the 8th term in the given expression.
$∵ T_{8} = T_{7 + 1}$

$∴ T_{8} = {}^{10}C_{7} (x^{3 / 2} y^{1 / 2})^{10 - 7} (- x^{1 / 2} y^{3 / 2})^{7} = - \frac{10 \times 9 \times 8}{3 \times 2} x^{9 / 2} y^{3 / 2} (x^{7 / 2} y^{21 / 2}) = - 120 x^{8} y^{12}$

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