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A right triangle $PQR$ is to be constructed in the $xy$-plane so that the right angle is at $P$ and line $PR$ is parallel to the $x$-axis. The $x$ and $y$ coordinates of $P$, $Q$ and $R$ are to be integers that satisfy the inequalities: $-4\le x\le5$ and $6\le y\le16$. How many different triangles could be constructed with these properties?

a) $110\quad $ b) $1, 100\quad $ c) $9, 900\quad $ d) $10, 000\quad $

My try:

$x=\{-4, -3, -2,\ldots 3, 4,5\}$ , $y=\{6, 7, 8,\ldots 14, 15,16\}$

there are 10 points on the x-axis & 11 points on the y-axis therefore

total number of triangles $=\binom{10}{2}\binom{11}{1}+\binom{10}{1}\binom{11}{2}=1045$

But my answer does not match any option. Please correct me if I am wrong. Somebody please help me solve it.
My book suggests that answer must be 9,900

thanks

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1 Answer 1

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Considering first the number of rectangles should help.

The number of rectangles is given by $$\binom{10}{2}\times\binom{11}{2}=2475$$ For each rectangle, we have four distinct right triangles, so the answer is $$4\times 2475=9900$$

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