It is a bit (or more than a bit tricky) to come up with a geometrical display of what is going on in cases more complicated than 4 people chosen 2 at a time... but here is a start for something a bit different which shows the data and the groups of people at each stage.
Manipulate[
Column[{
Row[{"Number of people:" , m}],
Row[{"All groups:" , -1 + 2^m - m}],
Grid[
{
{
Grid[
Join[
{{"Group size", "Number of groups"}},
Table[{i,
Tooltip[Binomial[m, i], Grid@Subsets[Range[m], {i}]]}, {i,
2, m}]
], Alignment -> Left,
Background -> {None, {set -> LightBlue}},
Dividers -> All, Spacings -> {1, 1}],
Spacer[20],
Grid@Subsets[Range[m], {set}]}
}, Alignment -> Top]
}
],
{{m, 2, "People"}, 2, 10, 1},
{{set, 2, "Set"}, 2, m, 1},
BaseStyle -> {11, FontFamily -> "Helvetica"}
]