coordinate geometry + combination

[armaan700+]

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 -6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?


A.54

B.432

C.2160

D.2916

E.148,824

[ajith]

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 -6 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions?


A.54

B.432

C.2160

D.2916

E.148,824

Is the answer C, if yes, I will post the explanation

[thephoenix]

dimensions of rectangle 9*6 (9 horizontal dots and 6 vertical).
AB is parallel to y-axis and AC is parallel to x-axis.
the (x,y) coordinates for vertex A can be chosen: 9C1*6C1;
the x coordinate for vertex C (as y coordinate is fixed by A) can be chosen: 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): 5C1, (6-1=5 as 1 vertical dot is already occupied by A).
9C1*6C*8C1*5C1=2160.
Answer: C.

[rahul.s]

dimensions of rectangle 9*6 (9 horizontal dots and 6 vertical).
AB is parallel to y-axis and AC is parallel to x-axis.
the (x,y) coordinates for vertex A can be chosen: 9C1*6C1;
the x coordinate for vertex C (as y coordinate is fixed by A) can be chosen: 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): 5C1, (6-1=5 as 1 vertical dot is already occupied by A).
9C1*6C*8C1*5C1=2160.
Answer: C.

i was starting at the problem for a few seconds trying to understand it. i think i finally understood the explanation. thanks.

[rahul.s]

armaan,

could you please mention the source of this problem?

[chintudave]

Answer is C.

Explanation:

a. The given matrix is 9 dots on x axis and 6 dots on y axis.
b. Choose 1 dot on y axis to plot the base of the triangle. 6C1 i.e 6
c. Choose 2 dots on x axis to plot the base on the y axis chosen in step b above. 9C2 i.e. 36
d. Now select 1 dot on y axis to plot the third vertex. But this should not be the same as the one in step b. Hence 5C1 i.e. 5
e. Now the x coordinates of the third vertex has to be one of the two dots in step c. 2C1 i.e. 2
f. So 6C1 * 9C2 * 5C1 * 2C1 = 6 * 36 * 5 * 2 = 2160.