What type of problem is finding the number of "alterations" on a vector whose components take values from a known set?

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What type of problem is finding the number of "alterations" on a vector whose components take values from a known set?

E.g.

$(x_0,x_1,x_2)$

$x_i \in \{0,1 \}$

Find all possible three-vectors.

Particularly, how should one "deal" with such vector problems? Should one e.g. list them as $x_0 x_1 x_2$ or something else, dismissing the indices?