How many words can be obtained using 2n letters?

Question

Given n letters 'A' and n letters 'B', how many unique words can be obtained using all 2n letters? Or how many unique arrangements exist for those letters? Why is the answer $C_{2n} ^n = \frac{2n!}{n!n!}$? I understood the fact that given 2n letters there are 2n! permutations and that because it's the same letter repeated there are identical permutations, but how do you reach that formula?

And in general, I've been told by my teacher than given a number of letters, the general formula would be $\frac{n!}{n_1! ... n_k!}$ where n is the total number of letters and $n_k$ the number of letters of the K-th letter. Why is that?

Answer 1

Let's assume for a moment that all $A'$s and $B'$s are distinct, say $A_1,A_2,\ldots A_n$ and same goes for the $B's$. One of the $2n!$ word formed by those letters would be $$ A_1A_2\cdots A_n B_1B_2\cdots B_n. \,\, (1) $$ At first, this word would be different than the word $$ A_nA_{n-1}\cdots A_1B_1B_2\cdots B_n, $$ but since in fact all the $A'$s are alike, all the words of the form $(1)$ are counted $n!$ times (fixing the $B'$s and permutting the $A's$. ) Since this is also true for the $B'$s, we get another $n!$ factor to divide our $2n!$ with.