# How many possible numbers do I have?

Stupid question from stupid non-math-orientated person here.

I have a list of four-digit sequences. These sequences consist of and iterate through a letter of the alphabet followed by a range of numbers from 100-999. So the list starts at A100, followed by A101, A102... A999, B100, B101... right up to Z999. Assuming each number in the list is unique and there are no repeats, how many permutations does that result in? How would I calculate it?

I had initially thought it was as simple as:

or

# $26 * 899 =$ 23,376 numbers

...but on looking deeper into the maths behind permutations and combinations I feel like I may have made a stupid assumption there. If I have and my initial calculation was wrong, how exactly would I go about doing this?

## 2 Answers

You are right except for the $899$! Note that $100$ is the first number. $101$ is the second, ..., $199$ is the 100th, ..., $999$ is the 900th!

So there are $26 \cdot 900$ items in the list.

• Damn, great catch. You also have no idea how glad I am that I'm not as bad at math as I thought, lol. Thanks so much. – Hashim Jan 10 '17 at 23:53
• @Hashim maths can be quite intuitive! As long as you use your brain to think, you won't do as bad as you think you will – RGS Jan 10 '17 at 23:56

Remember that the first number is $100$, so the $900^{th}$ number is 999.. You were right except this. so: Letters $(26)\times$ range $(900)$. $$26(900) = 23,400 \ \mathrm{solutions}$$

Keep in mind that you're range and total aren't the same thing. You have a total of $900$ numbers in each letter, even though you're range is 100-999 (inclusive) The generally proper way to express your range is 100-1000, as it's generally accepted that the last number is excluded from a set. This is also known as the fence post problem, explained here

• Your answer is incorrect, there are not 899 numbers but 900. – RGS Jan 10 '17 at 23:45
• @RSerrao fixed. I read the question, and still did it wrong myself. – Travis Jan 10 '17 at 23:46
• Haha, I'm glad to hear it wasn't just me. Thanks for the extra info about the range vs total. I also didn't know that about the last number being excluded - is there any particular reason for that? Seems like it could be confusing. – Hashim Jan 11 '17 at 0:02
• @Hashim, it makes the math part less confusing, $1000-100 = 900$ while $999-100 = 899$, which leads to the exact mistake you made. – Travis Jan 11 '17 at 0:04