【Question 23】
A positive integer, if from the first digit (the highest place) to a certain digit in the middle the digits are increasingly larger, and from this digit to the last digit (the unit place) the digits are increasingly smaller, is called a "convex number" (for example, 1357642 is a "convex number", while 7531, 1357462 are not "convex numbers"). How many nine-digit "convex numbers" can be formed using the digits 1-9?
【Analysis and Solution】
Counting.
Except for the digit 9, the other 8 digits must be arranged in descending order on either the left or right side of the digit 9; thus, there are $2^{8}=256$ ways to divide these 8 digits to the left or right of the digit 9;
However, the other 8 digits cannot all be placed on the left or right side of the digit 9;
Therefore, these nine digits can form a total of $256-2=254$ nine-digit “convex numbers”.