Question

使用数字 $1, 3, 5, 7$ 和 $8$ 各一次能组成的所有 $5$ 位数的平均数是多少？

What is the average (mean) of all $5$ -digit numbers that can be formed by using each of the digits $1, 3, 5, 7,$ and $8$ exactly once?

$48000$

$49999.5$

$53332.8$

$55555$

$56432.8$

答案：C

知识点：排列位值对称性

难度评级：1540

解答：

由对称性，五个数字在每个数位上出现的次数相同，所以每个数位上的平均数字为 $\dfrac{1 + 3 + 5 + 7 + 8}{5} = 4.8.$

因此平均数为 $\begin{gathered} \small 4.8(1 + 10 + 100 + 1000 + 10000) \\ = 4.8 \cdot 11111 = 53332.8. \end{gathered}$

所以正确答案是 C。

By symmetry, each of the five digits appears equally often in each place, so the average digit in every place is $\dfrac{1 + 3 + 5 + 7 + 8}{5} = 4.8.$

The average number is therefore $\begin{gathered} \small 4.8(1 + 10 + 100 + 1000 + 10000) \\ = 4.8 \cdot 11111 = 53332.8. \end{gathered}$

Thus, C is the correct answer.

2005 AMC 10B 第 20 题

答案：C

难度评级：1540

解答：

其他年份的第 20 题