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Subj:.....Inflationary Sequence (S649)
          From: MathNexus.wwu.edu
          on 5/13/2008
Drawing from MathNexus...

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Source: http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=166

At a mathematics conference recently, I picked up an 
interesting problems from one of the speakers (C.B., CWU). 

Problem: How many positive integers have the property 
that their digits increase as read from left to right? 
Some examples of such numbers are 19 or 356 or 12,679. 
 

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Hint: Some possibilities: 

Can any digits repeat? 

Try a smaller case...Say using the digits 1-5..., is 
there a pattern involved? 

Find an "intuitive" straight-forward way to solve the 
problem of all numbers less than a billion.

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Finger pointing down
from darrell94590 on 1/2/2006
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Drawing from Ripleys-Believe It Or Not
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              THE SOLUTION
 

Solution Commentary: Another hint of sorts....first write
down the special number 123456789.  Any number in our
sequence is "hidden" in this special number, just by 
crossing out some of the other digits. So, how many ways
can you chose a subset of the digits of this special
number...or how many ways to cross out some of the digits? 

Hope this enough to get you re-started on the problem....
 

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OK, here is my solution, based on their hint.

First, I will NOT deal with repeating digits and try the
   special number 12345.

Crossing out no digits from 12345 yields 1 number.

Crossing out one digits from 12345 yields 5 numbers.

Crossing out two digits from 12345 yields 10 numbers.
   I crossed out 12, 13, 14, 15, 23, 24, 25, 34, 35, and 45.

Crossing out three digits from 12345 yields 10 numbers.
   I crossed out 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345.

Crossing out four digits from 12345 yields 5 numbers.
   I crossed out 1234, 1235, 1245, 1345, and 2345.

This gives a total solution of 31 ways to the first five
digits and have the digits always increasing.
 

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Since this didn't reveal a pattern to me, I expanded the
search to sequences which contain 1, 2, 3, 4, and my 5 digits.

The sequence of 1 can be written 1 way.

The sequence of 12 can be written 3 way.

The sequence of 123 can be written 7 way.

The sequence of 1234 can be written 15 way.

And the sequence of 12345 can be written 31 way.

Each sequence is twice the preceeding sequence plus one.

So the sequence of 123456 can be written 63 way.

And the sequence of 1234567 can be written 127 way.

And the sequence of 12345678 can be written 255 way.

And finally, the sequence of 123456789 can be written 511 way.

So there are 511 ways to write a number less than a billion
with the digits always increasing, unless I made an error.

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