Tough one: Number of triangles

vk_vinayak wrote:A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordinates of A, B and C are to satisfy the inequalities -3 ≤ x ≤ 5 and 2 ≤ y ≤11 and x and y are integers. The number of different triangles that can be constructed with these properties are?

A. 90
B. 900
C. 6480
D. 8100
E. 10000

OA Later.

[spoiler]Is it necessary to assume that co-ordinates are integers? It would be helpful if somebody explained how to tackle this within 2 minutes.[/spoiler]

[spoiler]ans = c[/spoiler]

theCEO wrote:

vk_vinayak wrote:A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordinates of A, B and C are to satisfy the inequalities -3 ≤ x ≤ 5 and 2 ≤ y ≤11 and x and y are integers. The number of different triangles that can be constructed with these properties are?

A. 90
B. 900
C. 6480
D. 8100
E. 10000

OA Later.

[spoiler]Is it necessary to assume that co-ordinates are integers? It would be helpful if somebody explained how to tackle this within 2 minutes.[/spoiler]

[spoiler]ans = c[/spoiler]

It would have been helpful if you provided the solution.

theCEO wrote:

vk_vinayak wrote:A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordinates of A, B and C are to satisfy the inequalities -3 ≤ x ≤ 5 and 2 ≤ y ≤11 and x and y are integers. The number of different triangles that can be constructed with these properties are?

A. 90
B. 900
C. 6480
D. 8100
E. 10000

OA Later.

[spoiler]Is it necessary to assume that co-ordinates are integers? It would be helpful if somebody explained how to tackle this within 2 minutes.[/spoiler]

[spoiler]ans = c[/spoiler]

Just trying to solve....It may not be right

As we have two inequalities which bound the area under consideration.
The two inequalities are -3 ≤ x ≤ 5 and 2 ≤ y ≤11 which makes a square of 9(parallel to x-axis) * 10(parallel to y axis).

1. So we need to have AB || to x axis
so A can take x coordinate from these 9 points in 9C1 ways and also can take from 10C1 points from the y coordinate...


2. Now as AB is parallel to x-axis
so B can take 8 such another points as 1 horizontal point is already occupied by A in 8C1 ways.

and C can take 9 such points as 1 point is took by A in 9C1 ways...

so total triangles drawn will be = 10C1 * 9C1 * 8C1 * 9C1 = 10 * 9 * 8 * 9 = 6480