AMC10 2017 A

AMC10 2017 A · Q23

AMC10 2017 A · Q23. It mainly tests Combinations, Counting in geometry (lattice points).

How many triangles with positive area have all their vertices at points $(i, j)$ in the coordinate plane, where i and j are integers between 1 and 5, inclusive?

在坐标平面上有多少个具有正面积的三角形，其所有顶点位于点 $(i, j)$，其中 $i$ 和 $j$ 为 1 到 5 之间的整数（包含 1 和 5）？

(A) 2128 2128

(B) 2148 2148

(D) 2200 2200

(E) 2300 2300

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): There are $\binom{25}{3}=\frac{25\cdot24\cdot23}{6}=2300$ ways to choose three vertices, but in some cases they will fall on a line. There are $5\cdot\binom{5}{3}=50$ that fall on a horizontal line, another $50$ that fall on a vertical line, $\binom{5}{3}+2\binom{4}{3}+2\binom{3}{3}=20$ that fall on a line with slope $1$, another $20$ that fall on a line with slope $-1$, and $3$ each that fall on lines with slopes $2$, $-2$, $\frac{1}{2}$, and $-\frac{1}{2}$. Therefore the answer is $2300-50-50-20-20-12=2148$.

答案（B）：选择三个顶点共有 $\binom{25}{3}=\frac{25\cdot24\cdot23}{6}=2300$ 种方法，但有些情况下三点会共线。落在水平直线上的共有 $5\cdot\binom{5}{3}=50$ 种，落在竖直直线上的另有 $50$ 种；落在斜率为 $1$ 的直线上的共有 $\binom{5}{3}+2\binom{4}{3}+2\binom{3}{3}=20$ 种，落在斜率为 $-1$ 的直线上的另有 $20$ 种；此外，落在斜率为 $2$、$-2$、$\frac{1}{2}$、$-\frac{1}{2}$ 的直线上的各有 $3$ 种。因此答案为 $2300-50-50-20-20-12=2148$。

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