Combination with Repetitions Including Duplication of N Value Is a Combination with Repetition the correct term for the following problem


  N - Letters a, b, c
    R - 2


Example Result should equal


  aa
  ab
  ac
  bb
  ba
  bc
  cc
  ca
  cb  


Total Results: 9

However, in using every online calculator for combinations with repetitions I receive a total result of 6.

Example calculator: Combination with Repetition Calculator

What is the correct term for the mathematical equation / formula that I am looking for the accomplish the example data set provided from the variables presented?

Thanks!
Assuming you are not limited in the number of each letter which appears and you are asking how many ways you can have a total of $r$ letters appearing (with possible repitition) in a specific order (i.e. "baa" is considered different from "aba" and different from "aab"), this problem is often worded as

"Find the total number of strings of length $n$ from an alphabet with $r$ characters available."

Alternatively, one can think of this as "Find the number of functions from the set $\{1,2,\dots,n\}\to\{1,2,\dots,r\}$"

In either case, the result will be $r^n$.

This can be seen using multiplication principle.


Pick what the character in the first slot is (equivalently pick what $f(1)$ is).
Pick what the character in the second slot is
$\vdots$


In each step there are $r$ possibilities.  There are a total of $n$ steps.  Multiplication principle says that to get the total number of possibilities, we multiply the number of options at each step, giving a total of $r^n$



The calculator you link to answers a related but different question.  How many multisets of size $r$ exist with $n$ elements available to choose from?

Equivalently, how many integer solutions are there to the system $\begin{cases} x_1+x_2+\dots+x_n=r\\x_i\geq 0\end{cases}$

Equivalently, how many strings of length $r$ where letters must appear in alphabetical order exist taking characters from a set with $n$ characters available.

Equivalently, how many strings of length $r$ taking characters from a set with $n$ characters available exist where order of characters doesn't matter.  (i.e. $ab$ is considered the same as $ba$)

This can be seen via stars and bars to be $\binom{n+r-1}{r}$

In the example in the original post, the possibilities are aa, ab, ac, bb, bc, cc for a total of $6$ possibilities.  (ba, ca, and cb were not included since they are already in the list written as ab, ac, and bc respectively)
That is a permutation without repetition, not a combination.

n P r = n! / (n-r)! 

3 P 2 = 3! / (3-2)!

=3x2 / 1!

=6/1 = 6

Edit: I'm sorry didn't read your question properly, the correct term should be  "permutation with repetition" because the value A , B and C repeats eg. AA, BB, CC therefore it's n ^r.

3 ^2 
=3x3 =9 

It is a permutation because the order of the values matter example AB is different from BA. 

If it's a combination then the value of AB and BA will be the same because in combination order of the value doesn't matter u just combine them.