2007 Indonesia MO Problems/Problem 4
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.
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.
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