Find the coefficient of:

(vii) \[a^5 b^7\] in the expansion of \[\left( a - 2b \right)^{12}\]

Advertisement Remove all ads

#### Solution

Suppose *a*^{5} *b*^{7} occurs at the (*r* + 1)th term in the given expression.

Then, we have:

\[T_{r + 1} = ^{12}{}{C}_r a^{12 - r} ( - 2b )^r \]

\[ = ( - 1 )^r {12}{}{C}_r \left( a^{12 - r} \right) \left( b^r \right)\left( 2^r \right)\]

\[\text{ For this term to contain } a^5 b^7 , \text{ we must have: } \]

\[12 - r = 5 \]

\[ \Rightarrow r = 7\]

\[ \therefore \text{ Required coefficient } = ( - 1 )^7 {12}{}{C}_7 \left( 2^7 \right) = - \frac{12 \times 11 \times 10 \times 9 \times 8 \times 128}{5 \times 4 \times 3 \times 2} = - 101376\]

Concept: Introduction of Binomial Theorem

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads