# sequences and series – Combinatorics terminology for list of options

I am currently working with lists of options, where the options are numbered from $$1$$ to $$n_k$$ and are ordered increasingly, where $$k$$ is the number of lists.

Question 1: currently I am calling these simply sequences, do these have a more specific name?

From these list of options, I am generating $$k$$-tuples, where the elements of the tuples are the options of the lists, in the order the lists have been defined so they have the same index, like this:
$$S_1 = (1, 2, 3)$$
$$S_2 = (1, 2)$$
$$S_3 = (1, 2, 3, 4)$$
All $$k$$-tuples:
$$(1, 1, 1)$$
$$(1, 1, 2)$$
$$(1, 1, 3)$$
$$(1, 1, 4)$$
$$(1, 2, 1)$$
$$(1, 2, 2)$$
$$(1, 2, 3)$$
$$(1, 2, 4)$$
$$(2, 1, 1)$$
$$(2, 1, 2)$$
$$(2, 1, 3)$$
$$(2, 1, 4)$$
$$(2, 2, 1)$$
$$(2, 2, 2)$$
$$(2, 2, 3)$$
$$(2, 2, 4)$$
$$(3, 1, 1)$$
$$(3, 1, 2)$$
$$(3, 1, 3)$$
$$(3, 1, 4)$$
$$(3, 2, 1)$$
$$(3, 2, 2)$$
$$(3, 2, 3)$$
$$(3, 2, 4)$$
The order of the tuples doesn’t matter, so the number of $$k$$-tuples become $$n_1n_2…n_k$$, in the example it is $$3×2×4$$.

Question 2: currently I am calling this combinatorial operation simply all possibilities, does this have a specific name?