Find the coefficient of `a^5b^7` in `(a-2b)^(12)`

Explanation

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}(-2 b)^{r} `

`=(-1)^{r} 12 C_{r}(a^{12-r})(b^{r})(2^{r})`

For this term to contain `a^{5} b^{7}`, we must have:

`12-r=5 `

`Rightarrow r=7 `

`therefore` Required coefficient `=(-1)^{7}\ ^12 C_{7}(2^{7})=-frac{12 times 11 times 10 times 9 times 8 times 128}{5 times 4 times 3 times 2}=-101376`