1997 AMC 12 Problem 24

Question

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

Count how many rising numbers begin with

1

, then with

23

After those blocks, list the first few numbers beginning with

24

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.

Problem 24 in other years