Number Sequence Question: binary serial numbers on US dollar bills

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jasonking

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Aug 17, 2017
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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.
 
This is a wonderful exercise to help you practice your problem solving and pattern recognition skills. These skills will prove immensely helpful in all things math, going beyond just what's required in school or on the job, but even to fun problems like this. Often times it's easiest to start simple and build your way up to see if you can spot a pattern.

Let's start by assuming the serial numbers on the bills had just 2 digits. Well, it's easy to see that all the serial numbers would be "binary numbers." Not terribly interesting, but let's note it down anyway. How many possible numbers are there with two digits? As a hint, recall that you can pick any one of the ten digits for the first number and any one of the ten digits for the second number. Now, let's add a third digit and see what happens. The first two digits determine which digits are involved in the "binary number," so they can still be anything. Then the third digit must match one of these two numbers. So, with ten possibilities for the first digit, ten possibilities for the second digit, and two possibilities for the third digit, how many total "binary numbers" are there? Try adding a fourth digit and investigate what happens there. Are you finding a pattern? How many "binary numbers" would there be for a serial number with n digits?

"Ternary numbers" work the same way except now the first three digits are free to be any number, and only the fourth plus digits are constrained. Accordingly, how many "ternary numbers" are there with four digits? With five digits? With n digits? At this point, hopefully you've noticed an even larger pattern and can find "Quarternary numbers" which have exactly four unique digits, "Quinary numbers" which have exactly five unique digits, etc.
 
That is part of the problem I suppose I was having... the first number in the sequence on the low end is 00000001 - that number is a binary. All the way up to 11111111, when I have the first "solid serial" can I have a number with only two digits, as in this case the digit "0" is considered a digit for the purposes of labeling binary notes. I was thinking I may have to look at it from a top-down approach - 99999998 is the first binary when going backwards.


I will look at your suggestion, and see if I can deduce a pattern. Too early this morning. lol
 
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