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Question
The coefficient of x4 in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] is
Options
\[\frac{405}{256}\]
\[\frac{504}{259}\]
\[\frac{450}{263}\]
none of these
Solution
\[\frac{405}{256}\]
\[\text{ Suppose } x^4 \text{ occurs at the (r + 1)th term in the given expansion } . \]
\[\text{ Then, we have } \]
\[ T_{r + 1} = ^{10}{}{C}_r (\frac{x}{2} )^{10 - r} \left( \frac{- 3}{2 x^2} \right)^r \]
`= ( - 1 )^r " ^10C _r \frac{3^r}{2^{10 - r}} x^{10 - r - 2r} `
\[\text{ For this term to contain } x^4 , \text{ we must have: } \]
\[10 - 3r = 4\]
\[ \Rightarrow r = 2\]
\[ \therefore \text{ Required coefficient } = ^{10}{}{C}_2 \frac{3^2}{2^8} = \frac{10 \times 9 \times 9}{2 \times 2^8} = \frac{405}{256}\]
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