**Quote:**

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 coordiantes of A, B and C are to satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. The numbers of different triangles that can be constructed with these properties are?

A. 63

B. 336

C. 567

D. 3024

E. 5040

Here we have a rectangle of size 9*7.

There are 9 choices for the x coordinate {-1,0,1,2,3,4,5,6,7}

There are 7 choices for the y coordinate {1,2,3,4,5,6,7}

AB is parallel to the x-axis, so both A & B will have the same y-coordinate. Let us say that is p

AC is parallel to the y-axis, so both A & C will have the same x-coordinate. Let us say that is q

Let us assume the y coordinate of x coordinate of B to be r and the y-coordinate of C be s

So, the coordinates of A are (q,p); B are (r,p); and C are (q,s)

We need to find out overall possible values for p,q,r,s

q and r need to be chosen from the 9 choices for x-coordinate

=> q & r can be chosen in 9*8 = 72 ways

p and s need to be chosen from the 8 choices for y-coordinate

=> p & s can be chosen in 7*6 = 42 ways

Total ways = 72*42 = 3024

Correct Answer: D _________________

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