Calculating permutations when the set of elements contains "copies"

Question

Say we have a set of people with the same name:

S={A,B,C,C,D,E,F}

They are clearly,different people but two of them have the same name.

To use the permutations formula,it is demanded that the set contains different elements.Perhaps in this example,it shouldn't bother much,but in a situation where the set contains the characters:

S={a,a,b,b}

Can the formula be applied?If not,why?

By definition,sets contain different elements,is there any way to "distinct" the duplicate elements in order to use them at the formula?

Thanks!

Answer 1

If a multiset consists of $n_1,n_2,\dots n_k$ indistinguishable objects of distinguishable types $1,2\dots k$, the number of all distinct permutations of the objects is determined by the multinomial coefficient: $$\frac{n!}{\prod_i n_i!},$$ where $n=\sum_i n_i$ is the total number of objects in the multiset.