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I have this formula I ended up deriving after encountering couple of problems like this one, here it goes:
Setp 1: Make places for as many number of digits as the number of zero's after 1.
Setp 2: The first digit is the number of zeros after 1.
Setp 3: Then rest of the digits are zero.
Setp 4: Add one to this number if your upper range number is included in the count.
So total number of zero's between 1 and 1000, inclusive, is 301.
This count formula also works for all other digits except you don't do the Step #4. Of course, this formula has certain rules if your range doesn't end at a number that has 1 and tailing zeros. Which I'll mention some other time when I have more time at hand.
(# of zeros) * (Upper Range/10) [+ 1] = total number of 1's in the range
where [+ 1] is optional depending on your upper limit.