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I want to know how many combinations are possible with a 4 character 4 digit string. All capital. Ex. ABCD1234, AAAA0000 - ZZZZ9999.

What's the answer but more importantly, what's the formula?

Random attempt 1: 26+26+26+26+10+10+10+10 = 144

Random attempt 2: 26*26*26*26*10*10*10*10 = 4569760000
Rob Campbell
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  • Hi, Welcome to MSE. Users on this site like to answer those questions where the questioner put some effort by showing their work or adding their thoughts about the problem. As your post is lacking your attempt, there are possibilities that your question will attract some downvotes and may even be closed. To prevent that, please edit your question and provide context. Have a look at this post to know How to ask a good question on this site and what's the purpose of this site. –  Jun 08 '22 at 16:06
  • I understand but I don't know that I don't know. If I don't know the formula, I don't have any work to show. I could write something fairly silly like 26+26+26+26+10+10+10+10 = 144 which I know is way wrong. My thought is that someone would give the name of a formula or show the formula and I can then show my work to see if it is right but I don't know where to start. I thought starting by asking the question was the right thing to do here. Downvoting someone for not knowing the answer to the question their asking seems a bit counter-productive. 4569760000 (changed + to *) – Rob Campbell Jun 08 '22 at 16:24
  • Your "random attempt 2" you had $26\cdot 26\cdot 26\cdot 26\cdot 10\cdot 10\cdot 10\cdot 40$. Why did you end with a $40$ here instead of another $10$? – JMoravitz Jun 08 '22 at 16:27
  • The punchline is that the concept you are wanting to learn about is the Rule of Product and $26^4\cdot 10^4$ is the correct answer. – JMoravitz Jun 08 '22 at 16:28
  • It was supposed to be 10, it was a typo. Thanks. Guess that was a punchline huh :-). – Rob Campbell Jun 08 '22 at 18:02

1 Answers1

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The Rule of Product formally says that for finite sets $A,B$ we have that $|A\times B| = |A|\times |B|$

Informally in the context of combinatorics and counting problems, it can be paraphrased to say that if we want to count the number of possible outcomes a particular scenario has and every possible outcome can be uniquely described via a sequence of steps such that the number of options of how to continue at each step does not rely on what was chosen at earlier steps (though the list of available options may change) then the total number of outcomes is equal to the product of each of the number of options for each step.

In your case, you have $26$ options for the first letter. $26$ options for the second letter. Same for the third and fourth and then $10$ options for the first digit, $10$ options for the second digit and so on (including 10 options for the final digit, not 40).

The total number of possibilities then is $26\cdot 26\cdot 26\cdot 26\cdot 10\cdot 10\cdot 10\cdot 10$

To emphasize what I was talking about with "though the list of available options may change" consider the same problem but where we do not allow repeated letters or repeated digits. We have $26$ choices for the first letter, but after having made that choice (whatever choice that happened to be) we have only $25$ choices remaining available for the second letter. Which $25$ choices those were will depend on the first chosen letter, but regardless which it was it will remain $25$ choices. That problem has an answer of $26\cdot 25\cdot 24\cdot 23\cdot 10\cdot 9\cdot 8\cdot 7$

The Rule of Sum is another elementary counting principle which formally says that for finite sets who are disjoint $A,B$ (i.e. they have no overlapping elements) we have that $|A\cup B| = |A|+|B|$

N.B. When talking about overlapping sets, this leads to the Inclusion-Exclusion principle. More generally $|A\cup B|=|A|+|B|-|A\cap B|$

Informally, that is to say that if we want to count how many outcomes a particular scenario has and we can categorize these such that each outcome belongs only to a single category, we can count each category separately and then add the results together.

For example, if we have a class of boys and girls and we want to count how many total children there are we can simply count the boys first and add that amount to the number of girls.

JMoravitz
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