# Formula to calculate count of possible orderings (permutations) of a set of values

I'm working on a Google Sheets spreadsheet, and I'm having trouble figuring out a way to count all possible ways a set of (concrete) values could be sorted, taking into account possible duplicate values.

For example, if I have a list of 3 values `A,A,B`, I want to know how many ways I could shuffle them around. In this case the result would be (in no particular order):

``````A,A,B
A,B,A
B,A,A
``````

so 3 total combinations.

I am trying to figure out what formula I could use to input such a list and get the total combinations. I've tried to build formulas using `COMBIN`, `COMBINA`, `PERMUT` and `MULTINOMIAL`, combined with `COUNTUNIQUE` and `COUNTA`, but so far none of the results match what I expect (3).

Is this such an uncommon need? I thought it would be a well-known formula with many uses. e.g. to calculate the theoretical maximum of anagrams of a word, or counting the number of possible states of a shuffled deck of cards, etc. Maybe I'm missing the proper name for this operation?

I tried reading the Wikipedia articles on Binomial coefficient and Permutation, but they are quite hard to read.

I am comfortable building formulas on Google Sheets once I have a mental picture of the required steps, but what's frustrating me is that I can intuitively make sense of the problem and obtain the solution manually by exhaustively enumerating all the possibilities, but can't figure out how to convert that process into a generic mathematical equation. Any hints?

• Imagine A as Apple,B as Ball. In all 3sets,that you described, you end up with exactly 2 Apples and 1Ball. The number of combination is only 1. All 3 sets are not distinct. Is it permutation? Permutation,in general,does not allow repetition/duplicates(like A twice). This is more of a `n-tuple`.From a Set containing k elements(A,B),the number of possible n-tuples = `k^n` = 2^3 = 8. This `k^n` is only valid,Where no restrictions are placed on the repetition,i.e., (A,A,A),(B,B,A),(B,A,B),etc. are all possible. You're looking@ `n-tuples with restriction on repetition on both elements(A:2,B:1)` May 8, 2018 at 22:29
• You're better off asking MathOverflow. If you find a answer, Do update this post. May 8, 2018 at 22:30