Find the coefficient of

    Q2.    a^5b^7  in (a- 2b)^{12}

Answers (1)

As we know that the (r+1)^{th} term  T_{r+1} in the binomial expansion of  (a+b)^n  is given by 

T_{r+1}=^nC_ra^{n-r}b^r

Now let's assume a^5b^7 happens in the (r+1)^{th} term of the binomial expansion of (a- 2b)^{12}

So,

T_{r+1}=^{12}C_rx^{12-r}(-2b)^r

On comparing the indices of x we get,

r=7

Hence the coefficient of the  a^5b^7  in (a- 2b)^{12} is 

\\ \Rightarrow ^{12}C_7\times(-2)^7=\frac{12!}{5!7!}\times (-128)\\=\frac{12\times11\times10\times 9\times8}{5\times4\times3\times2}\times(-128) \\=-(729)(128) \\=-101376

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