Question

凸五边形 $ABCDE$ 的每个顶点都要指定一种颜色。有 $6$ 种颜色可选，并且每条对角线的两个端点必须颜色不同。共有多少种不同的涂色方法？

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?

$2520$

$2880$

$3120$

$3250$

$3750$

答案：C

知识点：图论容斥原理

难度评级：1820

解答：

对角线按顺序连接顶点 $A - C - E - B - D - A$ ，形成一个 $5$ -环。题目条件正是要求这个环被正常涂色。

长度为 $n$ 的环用 $k$ 种颜色作正常涂色的数量为 $(k-1)^n + (-1)^n (k-1)$ 。取 $n = 5$ 且 $k = 6$ ， $\begin{gathered} 5^5 + (-1)^5 \cdot 5 \\ = 3125 - 5 = 3120. \end{gathered}$

因此，正确答案是 C。

The diagonals connect the vertices in the order $A - C - E - B - D - A,$ which is a $5$ -cycle. The condition is exactly that this cycle is properly colored.

The number of proper $k$ -colorings of a cycle of length $n$ is $(k-1)^n + (-1)^n (k-1).$ With $n = 5$ and $k = 6,$ $\begin{gathered} 5^5 + (-1)^5 \cdot 5 \\ = 3125 - 5 = 3120. \end{gathered}$

Thus, the correct answer is C.

2011 AMC 12A 第 16 题

答案：C

难度评级：1820

解答：

其他年份的第 16 题