Real numbers $$x_1, x_2, \ldots, x_{2007}$$ are chosen such that $$(x_1, x_2), (x_2, x_3), \ldots, (x_{2006}, x_{2007})$$ are all points on the graph of $$y = \frac{1}{x - 1}$$.

A

Such a choice is possible for all real $$x_1 \ne 1$$

B

In every choice $$x_1, x_2, \ldots, x_{2007}$$ are all distinct

C

There are infinitely many choices in which all $$x_i$$'s are equal

D

There exists a choice in which the product $$x_2 x_3 \cdots x_{2007} = 0$$

E

None of these