Both Permutations/ Combination and Geometry (2 problems)

euro wrote: [2] 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 & 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

Official answer to [2] [spoiler](C) 6480 [/spoiler]

When a question asks for the number of triangles that can be constructed, it's not a geometry question but a combinations question. Why? Because a triangle is a combination of 3 points.

We need to determine how many ways we can combine A, B and C to form a triangle that satisfies all the requirements of the problem. For each point, we need to choose an x value and a y value.

Point A:
x value: -3≤x≤5, giving us 9 choices.

y value: 2≤y≤11, giving us 10 choices.

Now we have to combine the number of choices for x with the number of choices for y. It's as though we have 9 shirts and 10 ties, and we need to determine how many outfits can be made:

(number of choices for x)*(number of choices for y) = 9*10 = 90 choices for A.

Point C:
x value: In order to construct a right triangle, C has to have the same x coordinate as A (so that C is directly above A and we get a right angle). So we have only 1 choice for x: it must be the same integer that we chose for A's x value.

y value: If A and C share the same x value, they can't have the same y value, or they will be the same point. We used 1 of our 10 choices for y when we chose A, so we have 10-1 = 9 choices for B's y value.

(number of choices for x)*(number of choices for y) = 1*9 = 9 choices for C.

Point B:
y value: For AB to be parallel to the x axis, A and B have to share the same y value. So the number of choices for y is 1; it must be the same integer that we chose for A's y value.

x value: If A and B share the same y value, they can't have the same x value, or they will be the same point. We used 1 of our 9 choices for x when we chose A, so we have 9-1 = 8 choices for B's x value.

(number of choices for x)*(number of choices for y)= 8*1 = 8 choices for B.

So we have 90 choices for A, 9 choices for B, and 8 choices for B. We need to determine how many ways we can combine A, B and C to make a triangle. It's as though we have 90 shirts, 9 ties, and 8 pairs of pants, and we need to determine the number of outfits that can be made:

(number of choices for A)*(number of choices for B)*(number of choices for C) = 90*9*8 = 6480.

The correct answer is C.