2007 Indonesia MO Problems/Problem 4

Problem

A 10-digit arrangement is called beautiful if (i) when read left to right, form an increasing sequence, and form a decreasing sequence, and (ii) is not the leftmost digit. For example, is a beautiful arrangement. Determine the number of beautiful arrangements.

Solution

To start, the first digit of the arrangement must be a 9. For the next nine digits, five of them are in the set and four of them are in the set .

We can approach counting the number of beautiful arrangements by counting the number of ways to arrange 5 blue balls and 4 blue balls because by letting the blue balls represent elements in the set and letting the red balls represent elements in the set , there is only one way to order the elements in each set.

With this one-to-one correspondence, there are beautiful arrangements.

See Also

2007 Indonesia MO (Problems)
Preceded by Problem 3	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 5
All Indonesia MO Problems and Solutions