1997 AMC 12 第 24 题

Question

递增数是指每一位数字都大于其左边所有数字的正整数，例如

34689

。共有

\binom95=126

个五位递增数。将这些数从小到大排列后，列表中的第

97

个数不含数字

A rising number, such as

34689,

is a positive integer each digit of which is larger than each of the digits to its left. There are

\binom95=126

five-digit rising numbers. When these numbers are arranged from smallest to largest, the

97

th number in the list does not contain the digit

4

5

6

7

8

答案：B

知识点：组合 lexicographic order

难度评级：2120

小提示：

先数出以

1

开头的递增数有多少个，再数以

23

开头的

Count how many rising numbers begin with

1

, then with

23

大提示：

排除这两组后，列出最前面的几个以

24

开头的数

After those blocks, list the first few numbers beginning with

24

解答：

以

1

开头的共有

\binom84=70

个。在以

2

开头的数中，最前面的

\binom63=20

个以

23

开头，占据第

71

至第

90

位。因此总列表中的第

97

个数，是以

24

开头的第七个数：

\begin{gathered} 24567,24568,24569,24578,\\ 24579,24589,24678 \end{gathered}\text{。}

数

24678

不含

5

，所以正确答案是 B。

There are

\binom84=70

beginning with

1.

Among those beginning with

2,

the first

\binom63=20

begin with

23,

occupying positions

71

–

90.

Thus the

97

th overall is the seventh beginning with

24:

\begin{gathered} 24567,24568,24569,24578,\\ 24579,24589,24678. \end{gathered}

The number

24678

omits

5,

so B is correct.

1997 AMC 12 第 24 题

答案：B

难度评级：2120

小提示：

大提示：

解答：

其他年份的第 24 题