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?

Correct answer marked below

110

1,100

9,900

Correct

10,000

12,100

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Explanation

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Official Explanation

Official

We have the rectangle with dimensions

10 \times 11

(10 horizontal dots and 11 vertical).

PQ is parallel to y-axis and PR is parallel to x-axis.

Choose the (x,y) coordinates for vertex P (right angle):

10C1 \times 11C1

Choose the x coordinate for vertex R (as y coordinate is fixed by P):

9C1

(

10-1=9

as 1 horizontal dot is already occupied by P)…

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What this question tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.