# Find the coefficient of    Q2.    $a^5b^7$  in $(a- 2b)^{12}$

P Pankaj Sanodiya

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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