BMO1 2007 3

BMO1 2006 3

Solution:

So first let’s say that we have the arrangement _ 1 _ 2 _ 3 _ 4 _ 5 _

The _'s denote the spaces between the numbers “potential spaces for other numbers to go into 6-9”.

Now let’s see which spaces 6 can go into.
It can go before 1, between 1 and 2, between 2 and 3, between 3 and 4, and finally between 4 and 5.

If it’s after 5, then the digits 1-6 will be in order and that violates the problem condition.

Thus there are 5 choices for 6.

Now putting 6 in a spot gives way to one more possible slot.
For example, if we place 6 between 1 and 2 we get:

_ 1 _ 6 _ 2 _ 3 _ 4 _ 5 _

And thus we have 7 places to put 7. (note that this is two more than the number of slots for 6 since we can place 7 after 5)
Similarly, we have 8 places to put 8 and 9 places to put 9.

Multiplying, we get 2520 and we are done.

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