[Math] How many ways can we get 2 a’s and 2 b’s from aabb

Question

We have the following group: $aabb$

It is commutative, so abab is the same as aabb.

I have figured out this is a combinatorics question. Because abab is the same as aabb. I was how to solve these problems with the blank slot method, i.e. _ _ _ _.

If I do this manually, it's clear to me the answer is 6,

aabb
abab
abba
baba
bbaa
baab

Which is the same as $$\binom{4}{2}$$

But I don't really understand why this is true? How is this supposed to be done without brute forcing the question?

Answer 1

You have four slots. Choose two of them to be a, and the other two will be forced to be b. There are $\binom 4 2$ ways to choose two slots from four, giving the answer.