A right triangle ARC has to be constructed in the xy-plane so that the right angle is at A and AR is parallel to x axis. The coordinates of A, R and C are integers and satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. How many different triangles can be constructed with these properties? Choose the correct choice from the following options
A) 63	B) 336	C) 567	D) 3024	E) 5040
All the cordinates are integer hence,   possible AR values are   {-1,0}, {-1,1} ....{-1,7}: 8 ways   ....   {7,-1}, {7,0} ......{7,6}: 8 ways     9*8= 
<gadget id="calculator">  9*8</gadget>
<output>72</output>
72   for AC values can be   {1,2}, {1,3} .... {1,7} : 6ways   ......   {7,1}, {7,2} ..... {7,6} : 6 ways     7*6= 
<gadget id="calculator">  7*6</gadget>
<output>42</output>
42   Total= 72 * 42= 
<gadget id="calculator">72 * 42</gadget>
<output>3_024</output>
3024 hence 3024
<result>D</result>
A right triangle ARC has to be constructed in the xy-plane so that the right angle is at A and AR is parallel to x axis. The coordinates of A, R and C are integers and satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. How many different triangles can be constructed with these properties?