AMC8 2025

AMC8 2025 · Q25

AMC8 2025 · Q25. It mainly tests Basic counting (rules of product/sum), Counting & probability misc.

Makayla finds all the possible ways to draw a path in a $5 \times 5$ diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?

Makayla找出在$5 \times 5$菱形网格中绘制路径的所有可能方法。每条路径从网格底部开始，到顶部结束，总是向东北或西北移动一个单位。她计算每条路径与网格右侧之间的区域面积。下图显示了两个例子。所有可能路径确定的面积之和是多少？

(A) \ 2520 \ 2520

(B) \ 3150 \ 3150

(D) \ 4730 \ 4730

(E) \ 5050 \ 5050

Answer

Correct choice: (B)

正确答案：（B）

Solution

Step 1: To find the total number of paths, observe that all paths will have $10$ total steps. We have to choose which $5$ of these steps will be NE (the rest will be NW). So the total number of paths is $\binom{10}{5}$. The formula for combinations is: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ and $\binom{10}{5} = \frac{10!}{5!\times5!}=252$. Step 2: Each path splits the total area of $25$ in two parts. So, for any path that gives area = $A$, you can find a unique "sister" path that has an area = $25-A$ (in other words, the pair of paths have a combined area of 25). Possible ways to define the "sister" path are: - Rotate the entire grid $180^{\circ}$ - Swap each step of the original paths (for example, each NW becomes NE) (this is a reflection over the diagonal) Step 3: There are a few ways to get from this observation to the total area: - There are $252/2 = 126$ pairs of such paths, and the total area of each pair is $25$. So the total area given by all paths is $126 \times 25$. - Each of the $252$ paths gives an area of $25$ if you also count the "sister" paths. Since each "sister" path is also one of the $252$, you have to divide by $2$ to avoid double counting. So the total area given by all paths is $\frac{252 \times 25}{2}$. - Note that the average area of two "sister" paths is $\frac{25}{2}$, so you can think about every path having this area on average. So the total area given by all paths is $252 \times \frac{25}{2}$. The final answer is $\boxed{\textbf{(B)}~3150}.$ Note: This problem has a bijection (or 1-1 correspondence) , check out Intermediate Counting & Probability, Chapter 4, and Introduction to Counting & Probability, Chapter 5

步骤1：为了找到路径总数，观察所有路径将有$10$个总步数。我们必须选择这$10$步中的$5$步为NE（其余为NW）。因此路径总数是$\binom{10}{5}$。组合公式是：$\binom{n}{r} = \frac{n!}{r!(n-r)!}$ 且$\binom{10}{5} = \frac{10!}{5!\times5!}=252$。步骤2：每条路径将总面积25分成两部分。因此，对于给出面积=$A$的任意路径，你可以找到一个唯一的“姐妹”路径，其面积=$25-A$（换句话说，这对路径的组合面积为25）。定义“姐妹”路径的可能方法有： - 将整个网格旋转$180^{\circ}$ - 交换原始路径的每一步（例如，每个NW变为NE）（这是沿对角线的反射）步骤3：从这个观察到总面积有几种方法： - 有$252/2 = 126$对这样的路径，每对的总面积是$25$。因此所有路径的总面积是$126 \times 25$。 - 每个$252$路径如果也计入“姐妹”路径则给出面积$25$。由于每个“姐妹”路径也是$252$中的一个，你必须除以2以避免双重计数。因此所有路径的总面积是$\frac{252 \times 25}{2}$。 - 注意两条“姐妹”路径的平均面积是$\frac{25}{2}$，因此你可以认为每条路径平均有这个面积。因此所有路径的总面积是$252 \times \frac{25}{2}$。最终答案是$\boxed{\textbf{(B)}~3150}$。注意：此题有双射（或一一对应），参见中级计数与概率第4章，以及计数与概率导论第5章

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