Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates of A, B, and C are to be integers that are consistent with the inequalities -3 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?
Choices:
A) 54  B) 432  C) 2,160  D) 2,916  E) 148,824
We have the rectangle with dimensions 9*3 (9 horizontal dots and 3 vertical). AB is parallel to y-axis and AC is parallel to x-axis.   Choose the (x,y) coordinates for vertex A: 9C1*3C1;   Choose the x coordinate for vertex C (as y coordinate is fixed by A): 8C1, (9-1=8 as 1 horizontal dot is already occupied by A);   Choose the y coordinate for vertex B (as x coordinate is fixed by A): 2C1, (3-1=2 as 1 vertical dot is already occupied by A).   9C1*3C1*8C1*2C1=432.   Answer: 432.
<result>B</result>
Right triangle ABC is to be drawn in the xy-plane so that the right angle is at A and AB is parallel to the y-axis. If the x- and y-coordinates of A, B, and C are to be integers that are consistent with the inequalities -3 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?