# Question a6ac5

Jan 26, 2018

$- 101376$

#### Explanation:

Looking for the coefficient of ${a}^{5} {b}^{7}$ in the expansion of ${\left(a - 2 b\right)}^{12}$.

The binomial theorem says ${\left(x + y\right)}^{n} = {\sum}_{k = 0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) {x}^{n - k} \cdot {y}^{k}$

So ${\left(a - 2 b\right)}^{12} = {\sum}_{k = 0}^{12} \left(\begin{matrix}12 \\ k\end{matrix}\right) {a}^{12 - k} \cdot {\left(- 2 b\right)}^{k}$.

For the term we're seeking, we need the term when $k = 7$:

$\left(\begin{matrix}12 \\ 7\end{matrix}\right) {a}^{12 - 7} \cdot {\left(- 2 b\right)}^{7}$

$= 792 {a}^{5} {\left(- 2\right)}^{7} {b}^{7}$

$= - 792 \cdot 128 {a}^{5} {b}^{7}$

$= - 101376 {a}^{5} {b}^{7}$

so the coefficient is $- 101376$.

Note:
((12),(7))=(12!)/(7!(12-7)!) #
$= \frac{\left(12\right) \left(11\right) \left(10\right) \left(9\right) \left(8\right)}{\left(5\right) \left(4\right) \left(3\right) \left(2\right) \left(1\right)}$
$= \left(11\right) \left(9\right) \left(8\right) = 792$