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Question
The coefficient of \[x^{- 3}\] in the expansion of \[\left( x - \frac{m}{x} \right)^{11}\] is
Options
\[- 924 m^7\]
\[- 792 m^5\]
\[- 792 m^6\]
\[- 330 m^7\]
Solution
\[- 330 m^7\]
\[\text{ Let } x^{- 3} \text{ occur at (r + 1)th term in the given expansion } . \]
\[\text{ Then, we have} \]
\[ T_{r + 1} = ^{11}{}{C}_r x^{11 - r} \left( \frac{- m}{x} \right)^r \]
\[ = ( - 1 )^r \times ^{11}{}{C}_r m^r x^{11 - r - r} \]
\[\text{ For this term to contain } x^{- 3} , \text{ we must have } \]
\[11 - 2r = - 3\]
\[ \Rightarrow r = 7\]
`\therefore \text{ Required coefficient } = ( - 1 )^7 "^11C_7 m^7 `
\[ = - \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2} m^7 \]
\[ = - 330 m^7\]
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