Way to generate all multisets

Question

I have a set $\left\{\{1,2\}, \{3,4\}, \{5,6\}, \{7,8\}\right\}$ (for example) and want to generate all sets of $n$ elements with repetition but without counting the same multiset twice. So I want my generated list to contain: (say $n$ = 3)

$\{\{1,2\}, \{1,2\}, \{3,4\}\}$ and $\{\{1,2\}, \{3,4\}, \{5,6\}\}$ but I don't also want $\{\{1,2\}, \{3,4\}, \{1,2\}\}$ as that should be the same as the first set. (Well, I want this set but flattened, like $\{1,2,1,2,3,4\}$, etc, but I kept the brackets in to be more illustrative.)

Example: I have $\{\{1,2\}, \{3,4\}, \{5,6\}\}$ and $n =2$, then I want to generate $\{\{1,2,1,2\}, \{1,2,3,4\},\{1,2,5,6\}, \{3,4,3,4\}, \{3,4,5,6\}, \{5,6,5,5\}\} $

The only way I can think of is making a table then write some long loop thing to remove the repetitions and I'm sure there's a much more efficient way to do this. Probably invoke constant array somehow but I really have no idea how to implement it. Thanks for the help!

Answer 1

Your data:

data = Partition[Range[8], 2];

Read the documentation for Tuples as recommended by J.M and Prashanth.

Tuples[data, 3]

Generates a list of all possible 3-tuples of elements from list.

Because order is nor relevant you will have duplicates. So you need to sort and remove the permutations.

multisets[data_, n_] := Union[Sort /@ Tuples[data, n]]

Now use with n -> 3

multisets[data, 3]

{{{1, 2}, {1, 2}, {1, 2}}, {{1, 2}, {1, 2}, {3, 4}}, {{1, 2}, {1, 
   2}, {5, 6}}, {{1, 2}, {1, 2}, {7, 8}}, {{1, 2}, {3, 4}, {3, 
   4}}, {{1, 2}, {3, 4}, {5, 6}}, {{1, 2}, {3, 4}, {7, 8}}, {{1, 
   2}, {5, 6}, {5, 6}}, {{1, 2}, {5, 6}, {7, 8}}, {{1, 2}, {7, 8}, {7,
    8}}, {{3, 4}, {3, 4}, {3, 4}}, {{3, 4}, {3, 4}, {5, 6}}, {{3, 
   4}, {3, 4}, {7, 8}}, {{3, 4}, {5, 6}, {5, 6}}, {{3, 4}, {5, 6}, {7,
    8}}, {{3, 4}, {7, 8}, {7, 8}}, {{5, 6}, {5, 6}, {5, 6}}, {{5, 
   6}, {5, 6}, {7, 8}}, {{5, 6}, {7, 8}, {7, 8}}, {{7, 8}, {7, 8}, {7,
    8}}}