Problem Solving

Question :

Right triangle LMN is to be constructed in the xy-plane so that the right angle is at point L and LM is parallel to the x-axis. The x- and y- coordinates of L, M, and N are to be integers that satisfy the inequalities -3 =< x =< 4 and 3 =< y =< 11. How many different triangles with these properties could be constructed?


(A) 72

(B) 576

(C) 4032

(D) 4608

(E) 6336

OA - C

I was able to arrive at the correct answer using Method1.


However, I am missing something in Method2.

Method1 = Possibilities of "first" point in the plane = 9x8
Second point (on x-axis)=>any of the seven points = 7
Third point (on y-axis) = 8

Therefore, # = 9x8x7x8 = 4032.........................(A)

Method2: Let's twist the solution.

Choose two points on "x-axis" (I am ignoring "y coordinate"). Therefore, # of possibilities = 8x7 (Order matters because the right angle will change depending on which two points are chosen) .................(i)

Now for the y-coordinate,

X could be {-3 -2 -1 0 1 2 3 4}

Y could be {3 4 5 6 7 8 9 10 11}

We have a problem when x=3 or 4 and y = 3 or 4. (We don't want the y co-ordinate = x co-ordinate)

Therefore, if X lies in {-3 -2 -1 0 1 2}, the number of combinations = 6C1 x 9c1 (1 number is chosen from six "x-co od"s and 1 is chosen from "y-co od"s.

Similarly, if x lies in {3 4} then the number of combinations = 2c1 x 8c1 (we cannot choose x = y)

Total # of combinations = 9x6 + 2x8 = 54+16= 70...........(ii) {One could arrive at the same number of combinations if y is chosen first and then x is chosen}

I know that I am missing two combinations in (ii) so that (i)x(ii) = 9x8x8x7 i.e. the answer we derived from (A).

Any thoughts ?

Thanks