2011 AMC 12A 第 16 题

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

小提示：

五条对角线

AC, CE, EB, BD, DA

在这些顶点之间形成一个

5

-环

The five diagonals

AC, CE, EB, BD, DA

form a single

5

-cycle among the vertices

大提示：

长度为

n

的环用

k

种颜色作正常涂色的数量是

(k-1)^n + (-1)^n (k-1)

The number of proper colorings of a cycle of length

n

with

k

colors is

(k-1)^n + (-1)^n (k-1)

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}\text{。}

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

小提示：

大提示：

解答：

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