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Question
The coefficient of \[x^{- 17}\] in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] is
Options
1365
−1365
3003
−3003
Solution
−1365
\[\text { Suppose the (r + 1)th term in the given expansion contains the coefficient of } x^{- 17} . \]
\[\text{ Then, we have } : \]
\[ T_{r + 1} = ^{15}{}{C}_r ( x^4 )^{15 - r} \left( \frac{- 1}{x^3} \right)^r \]
` = ( - 1 )^r "^15C_r x^{60 - 4r - 3r} `
\[\text{ For this term to contain } x^{- 17} , \text{ we must have : \]
\[60 - 7r = - 17 \]
\[ \Rightarrow 7r = 77\]
\[ \Rightarrow r = 11\]
\[ \therefore \text{ Required coefficient } = ( - 1 )^{11}\]` "^ 15C_{11`=\[- \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2} = - 1365\]
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