AMC12 2010 A

AMC12 2010 A · Q18

AMC12 2010 A · Q18. It mainly tests Combinations, Coordinate geometry.

A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step?

一条 16 步路径从 $(-4,-4)$ 走到 $(4,4)$，每一步使 $x$ 坐标或 $y$ 坐标增加 1。有多少条这样的路径在每一步都保持在正方形 $-2 \le x \le 2$, $-2 \le y \le 2$ 的外部或边界上？

(A) 92 92

(B) 144 144

(D) 1698 1698

(E) 12800 12800

Answer

Correct choice: (D)

正确答案：（D）

Solution

Each path must go through either the second or the fourth quadrant. Each path that goes through the second quadrant must pass through exactly one of the points $(-4,4)$, $(-3,3)$, and $(-2,2)$. There is $1$ path of the first kind, ${8\choose 1}^2=64$ paths of the second kind, and ${8\choose 2}^2=28^2=784$ paths of the third type. Each path that goes through the fourth quadrant must pass through exactly one of the points $(4,-4)$, $(3,-3)$, and $(2,-2)$. Again, there is $1$ path of the first kind, ${8\choose 1}^2=64$ paths of the second kind, and ${8\choose 2}^2=28^2=784$ paths of the third type. Hence the total number of paths is $2(1+64+784) = \boxed{1698}$.

每条路径必须经过第二象限或第四象限。经过第二象限的每条路径必须且只会经过点 $(-4,4)$、$(-3,3)$、$(-2,2)$ 中的恰好一个。第一类有 $1$ 条路径，第二类有 ${8\choose 1}^2=64$ 条路径，第三类有 ${8\choose 2}^2=28^2=784$ 条路径。经过第四象限的每条路径必须且只会经过点 $(4,-4)$、$(3,-3)$、$(2,-2)$ 中的恰好一个。同样地，第一类有 $1$ 条路径，第二类有 ${8\choose 1}^2=64$ 条路径，第三类有 ${8\choose 2}^2=28^2=784$ 条路径。因此路径总数为 $2(1+64+784) = \boxed{1698}$。

Topics