Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x- and y-coordinates of P, Q and R are to be integers that satisfy the inequalities -4<=x<=5 and 6<=y<=16. How many different triangles with these properties could be constructed?  Choose the correct choice from the following options
A) 110  B) 1100  C) 9900  D) 10000  E) 12100
We know that P,Q, R can only lie on a grid 10x11   Let's start by fixing P' x-coordinate: we have 10 possible places for it   Then we choose x-coordinate for R: since PR || x-axis, R's x-coordinate can not coincide with P's x-coord, so we have only 9 possible choices left.   Then, we fix P's (and R's too) y-coordinate: 11 possible choices.   Finally, after we fixed P and R, Q's x-coordinate must be the same as P's (because P is the right angle and PR || x-axis), so the only choice is for y-coordinate, and we have 10 possible choices left.   So, total number of different triangles: 10*9*11*10=9,900   ANSWER:9900
<result>C</result>
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x- and y-coordinates of P, Q and R are to be integers that satisfy the inequalities -4<=x<=5 and 6<=y<=16. How many different triangles with these properties could be constructed?