5. On the $x y$-plane, find the number of triangles whose vertices have integer coordinates $(x, y)$ satisfying $1 \leq x \leq 4$ and $1 \leq y \leq 4$.
(1 mark)
在 $x y$ 平面上, 有些三角形的所有頂點 $(x, y)$ 皆是滿足 $1 \leq x \leq 4$ 和 $1 \leq y \leq 4$ 的整數。求這樣的三角形的數目。
5. 516
5. There are 16 ordered pairs $(x, y)$ of integers satisfying $1 \leq x \leq 4$ and $1 \leq y \leq 4$. Thus $C_{3}^{16}=560$ triangles can be formed. We must, however, delete those degenerate triangles, i.e. sets of three
points which are collinear. There are 4 horizontal lines of 4 points and 4 vertic al lines of 4 points. These together produce $(4+4) \times C_{3}^{4}=32$ degenerate triangles. Also, the points $(1,1)$; $(2,2) ;(3,3)$ and $(4,4)$ are collinear, leading to $C_{3}^{4}=4$ degenerate triangles, and so are the points $(1,4) ;(2,3) ;(3,2)$ and $(4,1)$. Finally, we also need to delete the 4 degenerate triangles $\{(1,2) ;(2,3) ;(3,4)\},\{(2,1) ;(3,2) ;(4,3)\},\{(1,3) ;(2,2) ;(3,1)\}$ and $\{(2,4) ;(3,3) ;(4,2)\}$. It follows that the answer is $560-32-4-4-4=516$.