Having a string of length $N$ with $M$ different letters, and knowing how many times each letter appears on the string, how can one calculate the amount of possible strings?
For example:
$N=10$
$M=2$
Characters $=\{A, B\}$
Amounts: $A=\{6, 4\}$
Possibilities:
AAAAAABBBB
BBBBAAAAAA
ABABABABAA
...
How can I calculate the amount of possibilities?
Edit, trying the answer provided in the comments
For these data:
$N=10$
$M=3$
Amounts $=\{2,5,3\}$
Would the solution be this one?
$$S={10\choose2}{8\choose5}{3\choose3}=45\times56\times1=2520$$
It looks like your solution is good. You have the right idea.
Now, if we calculate a bit more:
$${10 \choose 2}{8 \choose 5}{3 \choose 3} = \frac{10!}{2!8!}\frac{8!}{5!3!}\frac{3!}{3!0!} = \frac{10!}{2!5!3!}.$$
This is called a multinomial. Note that the factorials in the denominator match the numbers of each kind of letter you need. This holds in general, for any number of characters.