how to make a markov chain

Question

I have a list {{1,2,7},{3,4,6},{5,6,2}}. Then, I want to combinate it with itself so:

{{1,2,7},{3,4,6},{5,6,2}} X {{1,2,7},{3,4,6},{5,6,2}}  = newlist= 
{{{1,2,7},{1,2,7}},{{1,2,7},{3,4,6}},{{1,2,7},{5,6,2}},{{3,4,6}, 
{1,2,7}},{{3,4,6},{3,4,6}}......} 

Then I want to apply the next function: (i am gonna make the example with the second element of the newlist)

 (N!/1!2!7!)*(3/3+4+6)^1*(4/3+4+6)^2*(6/3+4+6)^7

Answer 1

It's not clear in the question but from the context I assume you are trying to build a transition matrix.

Assuming that N! was intended to read 3! (where presumably N=3, the dimension of each list) (see N) the following code evaluates to a $3\times 3$ matrix which will subsequently become a transition matrix (in fact, it can be done simultaneously, but I figured it would be better to present the intermediate steps)

(* set up the input *)
x = {{1, 2, 7}, {3, 4, 6}, {5, 6, 2}};

(* produce the matrix and display it *)
y = Outer[f, x, x, 1]

y // MatrixForm

The following function will transform the entries of the preceding matrix into numerical quantities

With[{n = 3!},
  (* note how Factorial threads over lists *)
  f = n/Times @@ Factorial[#1] Times @@ ((#2/Plus @@ #2)^#1) &;
 ]

Now, evaluating

y // N // Chop // MatrixForm

will produce the corresponding transition matrix:

Having said that, it is obvious that the output matrix is not a transition matrix (rows do not sum to 1); this means that my original assumption of what your shorthand calculations were trying to achieve was wrong;

Perhaps more context can be provided; also taking the cue from the question title, DiscreteMarkovProcess might be a relevant resource for you.