Is there any solution on the Wolfram offering of Platforms that would allow me to create a blog of sorts (or some type of personal portal) as I figure out mathematical concepts of a model that I am building? For instance, the question I posed above is just one part of a greater project that I have spent much of my life trying to explain, and only now am I turning to Mathematica to hopefully prove a new information system model. I understand this quest is probably completely out of my league but before I could talk or read, I worked on it the majority of my life and only now realize it just may be possible to prove my model using Mathematica. I am attempting to prove each of these concepts using the Wolfram community as a means to help find an easy way to demonstrate them. However, there are hundreds of these concepts that build upon one another to form a universal system model as I attempt to prove the link between linguistics, geometry, and mathematics in explaining the model.

As I ask these types of questions of the Wolfram community, I hope to gain a better mathematical understanding of each of these concepts as I attempt to build an overarching theory and thus a possible proof of concept for a different method of designing a behaviorally governed information system. So, my question is: What would be the best way to share these concepts interactively while I explain to my audience why each of these concepts are important in the model we are attempting to prove?

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"}

 ]

It's equivalent to examining integers in base 2, of length 4 (or n, or whatever). Use a digit to denote each individual. Set it to 1 if that individual is in a group and zero if not. So each integer corresponds to a distinct group. How many integers can you form in this way?

For the restricted case, how many such integers have exactly 2 or 3 ones? However you decide to demonstrate this, you will be hard pressed to avoid the binomial (as in "n-choose-k") function.

Thank you for your Response David Reiss, and to be even more specific and to allow an audience to be interactive; How would I program it to use a slide bar starting with one through ten and it show in the form of some graphical representation, with the correlating numbers for each group. For example, using four people I need it to demonstrate both graphically as well as real number representation that four people create 4 triads (Groups of three) and 6 diads / pairs (Groups of two), thus totaling 10 groups, plus the one group of four, which is then of course a total of eleven.

Although this might not be the correct terminology, the way I currently demonstrate this on lets say a napkin :), I draw four circles in the form of a square, each representing a single person, then by circling each pair I demonstrate that there are 6 possible diads (pairs), and then continuing the demonstration I circle four sets of three which shows a possibility of four triads, or groups of three. Then at the end, I circle all four to demonstrate the fact that four is the last and final group created.

My goal is to find a solution that replaces the napkin demonstration, hopefully in the form of a fun tool for even a kindergarten level.

Assuming that you mean groups that containing always more than one person this is given in general by (your initial case is m=4):

Sum[Binomial[m, i], {i, 2, m}]

which gives

-1 + 2^m - m

and you can visualize its growth with m

ListPlot[Table[-1 + 2^m - m, {m, 2, 10}]]