Werk $$$\left(2 x - 3 y\right)^{7}$$$ uit

Werk $$$\left(2 x - 3 y\right)^{7}$$$ uit.

De ontwikkeling wordt gegeven door de volgende formule: $$$\left(a + b\right)^{n} = \sum_{k=0}^{n} {\binom{n}{k}} a^{n - k} b^{k}$$$, waarbij $$${\binom{n}{k}} = \frac{n!}{\left(n - k\right)! k!}$$$ en $$$n! = 1 \cdot 2 \cdot \ldots \cdot n$$$.

We hebben dat $$$a = 2 x$$$, $$$b = - 3 y$$$ en $$$n = 7$$$.

Daarom geldt $$$\left(2 x - 3 y\right)^{7} = \sum_{k=0}^{7} {\binom{7}{k}} \left(2 x\right)^{7 - k} \left(- 3 y\right)^{k}$$$.

Bereken nu het product voor elke waarde van $$$k$$$ van $$$0$$$ tot en met $$$7$$$.

$$$k = 0$$$: $$${\binom{7}{0}} \left(2 x\right)^{7 - 0} \left(- 3 y\right)^{0} = \frac{7!}{\left(7 - 0\right)! 0!} \left(2 x\right)^{7 - 0} \left(- 3 y\right)^{0} = 128 x^{7}$$$

$$$k = 1$$$: $$${\binom{7}{1}} \left(2 x\right)^{7 - 1} \left(- 3 y\right)^{1} = \frac{7!}{\left(7 - 1\right)! 1!} \left(2 x\right)^{7 - 1} \left(- 3 y\right)^{1} = - 1344 x^{6} y$$$

$$$k = 2$$$: $$${\binom{7}{2}} \left(2 x\right)^{7 - 2} \left(- 3 y\right)^{2} = \frac{7!}{\left(7 - 2\right)! 2!} \left(2 x\right)^{7 - 2} \left(- 3 y\right)^{2} = 6048 x^{5} y^{2}$$$

$$$k = 3$$$: $$${\binom{7}{3}} \left(2 x\right)^{7 - 3} \left(- 3 y\right)^{3} = \frac{7!}{\left(7 - 3\right)! 3!} \left(2 x\right)^{7 - 3} \left(- 3 y\right)^{3} = - 15120 x^{4} y^{3}$$$

$$$k = 4$$$: $$${\binom{7}{4}} \left(2 x\right)^{7 - 4} \left(- 3 y\right)^{4} = \frac{7!}{\left(7 - 4\right)! 4!} \left(2 x\right)^{7 - 4} \left(- 3 y\right)^{4} = 22680 x^{3} y^{4}$$$

$$$k = 5$$$: $$${\binom{7}{5}} \left(2 x\right)^{7 - 5} \left(- 3 y\right)^{5} = \frac{7!}{\left(7 - 5\right)! 5!} \left(2 x\right)^{7 - 5} \left(- 3 y\right)^{5} = - 20412 x^{2} y^{5}$$$

$$$k = 6$$$: $$${\binom{7}{6}} \left(2 x\right)^{7 - 6} \left(- 3 y\right)^{6} = \frac{7!}{\left(7 - 6\right)! 6!} \left(2 x\right)^{7 - 6} \left(- 3 y\right)^{6} = 10206 x y^{6}$$$

$$$k = 7$$$: $$${\binom{7}{7}} \left(2 x\right)^{7 - 7} \left(- 3 y\right)^{7} = \frac{7!}{\left(7 - 7\right)! 7!} \left(2 x\right)^{7 - 7} \left(- 3 y\right)^{7} = - 2187 y^{7}$$$

Dus, $$$\left(2 x - 3 y\right)^{7} = 128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 10206 x y^{6} - 2187 y^{7}.$$$

$$$\left(2 x - 3 y\right)^{7} = 128 x^{7} - 1344 x^{6} y + 6048 x^{5} y^{2} - 15120 x^{4} y^{3} + 22680 x^{3} y^{4} - 20412 x^{2} y^{5} + 10206 x y^{6} - 2187 y^{7}$$$A