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Question
Find the coefficient of:
(vii) \[a^5 b^7\] in the expansion of \[\left( a - 2b \right)^{12}\]
Solution
Suppose a5 b7 occurs at the (r + 1)th term in the given expression.
Then, we have:
\[T_{r + 1} = ^{12}{}{C}_r a^{12 - r} ( - 2b )^r \]
\[ = ( - 1 )^r {12}{}{C}_r \left( a^{12 - r} \right) \left( b^r \right)\left( 2^r \right)\]
\[\text{ For this term to contain } a^5 b^7 , \text{ we must have: } \]
\[12 - r = 5 \]
\[ \Rightarrow r = 7\]
\[ \therefore \text{ Required coefficient } = ( - 1 )^7 {12}{}{C}_7 \left( 2^7 \right) = - \frac{12 \times 11 \times 10 \times 9 \times 8 \times 128}{5 \times 4 \times 3 \times 2} = - 101376\]
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