Question 945583: A book has 500 pages numbered 1, 2, 3… and so on. How many times does the digit 1 appear in the page numbers?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The digit 1 sometimes appears more than once on the same page number,
so it appears a lot.
It appears at  times as the first digit in pages 100 to 199.
It appears as the last digit  times
(on page 1, and in other page numbers preceded by 1, 2, 3, ... 49).
It appears as the tens digit in  2-digit numbers (10, 11, ...19),
and also appears as the tens digit in many 3-digit page numbers,
times preceded by each of the  digits 1, 2, 3,and 4.
So, the total count is
 appearances of the digit 1.
NOTE:
That does not mean that there are 200 page numbers where the number 1 appears.
That would be a different question.
In that case, I would rename the page numbers as the 3-digit numbers
001, 002, 003, .... 498, 499, 000.
(I am using 001 for 1, 002 for 2, and so on up to 099 for 99, and I use 000 for 500).
Those are 500 3-digit ordered arrangements,
with 5 choices for the first digit,
and 10 choices for each of the other two digits.
There is a total of  such arrangements.
How many contain the digit 1 at least once?
How many do not contain y=the digit 1 at all?
If we wanted to exclude the digit 1,
we could only use  of the  arrangements.
so  pages contain the digit 1 at least once.
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