Background to my problem:

I collect fancy serial numbers that are on US Dollars. I have been writing a guide for other collectors, and I need to calculate the number of a pedigree of serial number. A pedigree is what a fancy serial is labeled. In this problem, I need to calculate the number of "binary serial numbers" in a run of bills. A run can start with the serial number 00000001 and end with 99999999. A binary serial number is a serial number that only uses 1's and 0's. So, for example, 10100110 is a "binary serial number.

Basically, I need the formula for calculating the total number of binary serial numbers.

I also need to know if the formula would be accurate of other digits used in the same set of numbers. For example, 23322233 would also be a binary number. I also need to know how many combinations only use 2 and 3.

So, I need to solve for 0s and 1s only, and then the combinations for all the other possible combinations (e.g. 0-2, 0-3, 04, 0-5, 0-6, 0-7, 0-8, 0-9, 1-2, 1-3 ... ect).

After that, I need to do trinary serials in the same set.. for example the serial number 12312312 is a Trinary serial.

I know that has to be a way to solve this with mathematics. Thanks for your consideration.

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