If with the Binomial Coefficient we try to find the possible combinations $\binom{n}{k}$ where $n$ is equal to $k$ what is the theory behind factorial resulting in the correct solution?
E.g. $\binom{4}{4} = 4! = 4\times3\times2\times1 = (4+4+4)+(4+4+4) = 24$
Note, that I set $n$ to equal $k$ for simplicity of the question. What I'm struggling to understand is why multiplication (with factorial) is able to provide a method to compute the different combinations (what is the meaning of $4\times3\times2\times1$ when creating combinations).
The combinatorial argument is that to imagine finding ${n \choose k}$ you can think of lining up $n$ objects in a row. You can do that in $n!$ ways-do you know that? Then to choose $k$ of them, any of the $k!$ orderings of the first $k$ items are equivalent, as are any of the $(n-k)!$ orderings of the last $n-k$ items. So the distinguishable cases are ${n \choose k}=\frac {n!}{k!(n-k)!}$