I know that visually different means that for example for the set AABB the 2 As are treated the same so $A_1A_2B_2B_1$ is the same as $A_1A_2B_1B_2$ but i am unsure how to express this mathematically, in particular how to calculate it for more complex examples such as DISCRETEMATHEMATICS where there are $11$ unique letters and $19$ slots for them, i think it is something along the lines of $\frac{11!}{19!(11-19)!}$ but this gives me a negative factorial $(-8)!$.
Any help is appreciated
On
First assume that all the letters are different. This gives you $19!$ different arrangements. And then start to take account that some are indifferentiable. In your case you have $2$ $A$s, in how many ways can these to be arranged? The answer is $2!$ so this has to be divided out since the exchange of $A$ and $A$ does not give rise to a new arrangement. Further you have $2$ $C$s and these can be arranged in $2!$ different ways so we have to divide this out as well. Further you have $3$ $E$s these can be arraned in $3!$ different ways, and so on. Finally you will arrive at $$ \frac{19!}{2!2!3!2!2!2!3!}. $$
Sorting the letters we get
AACCDEEEHIIMMRSSTTTSo it should be
$$\binom{19}{2,2,1,3,1,2,2,1,2,3} = \frac{19!}{2!2!1!3!1!2!2!1!2!3!}$$
The notation on the left is the Multinomial Coefficient.