Expand $$$\left(2 x - 3 y\right)^{6}$$$

Expand $$$\left(2 x - 3 y\right)^{6}$$$.

The expansion is given by the following formula: $$$\left(a + b\right)^{n} = \sum_{k=0}^{n} {\binom{n}{k}} a^{n - k} b^{k}$$$, where $$${\binom{n}{k}} = \frac{n!}{\left(n - k\right)! k!}$$$ and $$$n! = 1 \cdot 2 \cdot \ldots \cdot n$$$.

We have that $$$a = 2 x$$$, $$$b = - 3 y$$$, and $$$n = 6$$$.

Therefore, $$$\left(2 x - 3 y\right)^{6} = \sum_{k=0}^{6} {\binom{6}{k}} \left(2 x\right)^{6 - k} \left(- 3 y\right)^{k}$$$.

Now, calculate the product for every value of $$$k$$$ from $$$0$$$ to $$$6$$$.

$$$k = 0$$$: $$${\binom{6}{0}} \left(2 x\right)^{6 - 0} \left(- 3 y\right)^{0} = \frac{6!}{\left(6 - 0\right)! 0!} \left(2 x\right)^{6 - 0} \left(- 3 y\right)^{0} = 64 x^{6}$$$

$$$k = 1$$$: $$${\binom{6}{1}} \left(2 x\right)^{6 - 1} \left(- 3 y\right)^{1} = \frac{6!}{\left(6 - 1\right)! 1!} \left(2 x\right)^{6 - 1} \left(- 3 y\right)^{1} = - 576 x^{5} y$$$

$$$k = 2$$$: $$${\binom{6}{2}} \left(2 x\right)^{6 - 2} \left(- 3 y\right)^{2} = \frac{6!}{\left(6 - 2\right)! 2!} \left(2 x\right)^{6 - 2} \left(- 3 y\right)^{2} = 2160 x^{4} y^{2}$$$

$$$k = 3$$$: $$${\binom{6}{3}} \left(2 x\right)^{6 - 3} \left(- 3 y\right)^{3} = \frac{6!}{\left(6 - 3\right)! 3!} \left(2 x\right)^{6 - 3} \left(- 3 y\right)^{3} = - 4320 x^{3} y^{3}$$$

$$$k = 4$$$: $$${\binom{6}{4}} \left(2 x\right)^{6 - 4} \left(- 3 y\right)^{4} = \frac{6!}{\left(6 - 4\right)! 4!} \left(2 x\right)^{6 - 4} \left(- 3 y\right)^{4} = 4860 x^{2} y^{4}$$$

$$$k = 5$$$: $$${\binom{6}{5}} \left(2 x\right)^{6 - 5} \left(- 3 y\right)^{5} = \frac{6!}{\left(6 - 5\right)! 5!} \left(2 x\right)^{6 - 5} \left(- 3 y\right)^{5} = - 2916 x y^{5}$$$

$$$k = 6$$$: $$${\binom{6}{6}} \left(2 x\right)^{6 - 6} \left(- 3 y\right)^{6} = \frac{6!}{\left(6 - 6\right)! 6!} \left(2 x\right)^{6 - 6} \left(- 3 y\right)^{6} = 729 y^{6}$$$

Thus, $$$\left(2 x - 3 y\right)^{6} = 64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} - 2916 x y^{5} + 729 y^{6}.$$$

$$$\left(2 x - 3 y\right)^{6} = 64 x^{6} - 576 x^{5} y + 2160 x^{4} y^{2} - 4320 x^{3} y^{3} + 4860 x^{2} y^{4} - 2916 x y^{5} + 729 y^{6}$$$A