How many possibilities permutation Let us say, I have the following: DADA which we can assign an index to $D_1$ $A_1$ $D_2$ $A_2$.
The possibilities, if the order of every letter in this word counts:
$D_1$ $A_1$ $D_2$ $A_2$
$D_1$ $D_2$ $A_1$ $A_2$
$D_1$ $A_1$ $A_2$ $D_2$
$A_1$ $A_2$ $D_1$ $D_2$
$A_1$ $D_1$ $D_2$ $A_2$
I get 5 possibilities. However, if I take the following formula:
$$\frac{n!}{n_D!\cdot n_A!}$$
plugging in the following information: $n=4$, $n_D=2$ and $n_A=2$, returns a 6, which shows that there's one more possibility or combination.
Is this correct? Also, how would this be, if the order of every letter didn't count, and instead this forms whole words.
EDIT: Every letter with a unique index is treated as a unique single letter. 


Is this the way the formula should be used?

 A: 
The possibilities, if the order of every letter in this word counts

As you have said, if the order of every letter matters, then you are supposed so treat each letter as a completely different letter (i.e., D1 is different from D3).
So, So, Number of ways you can arrange 4 completely different elements, keeping each at one of the four given places, without repetition of any element, is given by nPr = n!/(n-r)!
So, 4!/(4-4)! = 4!/0! = 4! = 24 ways, i.e, you can arrange them in 24 different ways.

If you want to consider both D1 and D3 as same, then the number of ways you can arrange those letters is given by nPr/nD = n!/(n-r)!•nD.
Similarly, if you also want to consider A2 and A4 as same letters, then number of ways you can arrange those letters is given by nPr/nD•nA = n!/(n-r)!•nD•nA.

The problem with your answer is that you're are considering all the repeating letters as different and trying to find their permutations, but in calculation, you're using the formula used to find the number of possibilities when repeating letters are NOT treated as different letters
A: The given word is DADA, it is a four letter word, but two letters, D and A, are repeated twice.
Let's, initially for no reason (don't worry about that), assign indices to all four letters. In this way: D1A1D2A2.
CASE-1
Let's consider D1 and D2 are not the same letter (element), similarly D1 and D1 are not same letters, they all are different from each other.
i.e., D1 ≠ D2 and A1 ≠ A2.
Now, let's list all the ways in which we can arrange these letters -

*

*A1 A2 D1 D2


*A1 A2 D2 D1


*A2 A1 D1 D2


*A2 A1 D2 D1
--


*A1 D1 A2 D2


*A1 D2 A1 D2


*A2 D1 A2 D1


*A2 D2 A1 D1
--


*D1 A1 A2 D2


*D1 A2 A1 D2


*D2 A1 A2 D1


*D2 A2 A1 D1
--


*D1 A1 D2 A2


*D1 A2 D1 A2


*D2 A1 D2 A1


*D2 A2 D1 A1
--


*D1 D1 A2 A2


*D1 D2 A1 A2


*D2 D1 A2 A1


*D2 D2 A1 A1
--


*A1 D1 D2 A2


*A1 D2 D1 A2


*A2 D1 D2 A1


*A2 D2 D1 A1
You can see that, by considering D1 and D2 as different letters and A1 and A2 as different letters, we get a total of 24 different ways of arrangements.

CASE-2
Now, let's consider D1 and D2 as the same letters and A1 and A2 also as the same letters (you can erase the indices), i.e., D1 = D2 = D, and A1 = A2 = A.
So, let's again list those 24 permutations, but now, erasing those indices:

*

*A A D D


*A A D D


*A A D D


*A A D D
--


*A D A D


*A D A D


*A D A D


*A D A D
--


*D A A D


*D A A D


*D A A D


*D A A D
--


*D A D A


*D A D A


*D A D A


*D A D A
--


*D D A A


*D D A A


*D D A A


*D D A A
--


*A D D A


*A D D A


*A D D A


*A D D A
You can see that by erasing those indices, you're list same type of arrangement 4 times.
So, if you not consider repeated letters as different letters, then number of different permutations are 24/4 = 6 (since there are 4 types of same arrangement).
Edit 1{
The formula that you were using: n!/nD!⋅nA!, is applicable here, where nD = 2 (number of D's) and nA = 2 (number of A's). So nD!⋅nA! = 2!.2! = 4, which is exactly exactly equal to what we are dividing from n!, i.e., 24/4 = 6.
}
So, let's list them again by ignoring repeated arrangements-

*

*A A D D


*A D A D


*D A A D


*D A D A


*D D A A


*A D D A
So, hence 6 permutations, if you not consider repeated letters as different letters.

Please consider reading my both answers for maximum details.
