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How many groups can be formed, using four people?

This is a question I ask many people, and in my experience most do not know the answer. "How many groups can be formed, using four people?" As someone new to the Wolfram Language and Mathematica, could someone help me figure out how I could "demonstrate" the answer to this question, using Mathematica? In this case, I already know the answer to this question is ten, or eleven if you count the four as a single group. However, I would like to not only demonstrate this, but how it incrementally increases with every additional person introduced.

POSTED BY: Brian Woytovich
11 Replies

One approach to blog production is simply to use WordPress. The issue with that is that, if you want to show a lot of Mathematica code in it--formatted, then that may require some detailed work to get any formatting right. However, if you want to, for example, embed CDFs in a WordPress document, then that can be done using the CDF plugin for WordPress (it can be found by searching for the plugin in WordPress).

Wolfram Research internally clearly has some tools for blog production which is used to create it's blog posting, but I do not know the degree to which that is something that they will eventually offer as a consumer tool.

I wrote some blogging tools a long while ago which are usable but do have a learning curve and need one ot have one's own web hosted URL (an old example of its use is here: http://www.scientificarts.com/worklife/wlfwblog/index.html )

But if you have access to a web server where you can put up web pages a very basic way to offer what you write in a Mathematica notebook as a static web page representation of that notebook is to use the Export function to export the notebook as HTML as documented here:

http://reference.wolfram.com/language/ref/format/HTML.html

POSTED BY: David Reiss

This is not an answer to your question of blog production although I believe there are useful tools that others might mention. For ideas on use of Mathematica in illustrating concepts of myriad sorts, you might want to have a look at

demonstrations.wolfram.com

Suffice it to say that many areas are represented.

A note on terminology: typically one speaks (writes) of model validation rather than proving. I don't know what it would mean to "prove" a model, but "validating" generally means "showing it makes sense" e.g. by experiment.

POSTED BY: Daniel Lichtblau

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?

POSTED BY: Brian Woytovich

Daniel Lichtblau, I will be sure to read up on this topic more extensively and continue to explore alternate methods to utilize this function. Thank you!

POSTED BY: Brian Woytovich

Oh wow, this is great David Reiss! Let me take a look at this some more. Thanks!

enter image description here

POSTED BY: Brian Woytovich

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"}

 ]
POSTED BY: David Reiss

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.

POSTED BY: Daniel Lichtblau

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.

POSTED BY: Brian Woytovich

The question was about the total:

In[1]:= -1 + 2^m - m /. m -> 4

Out[1]= 11

The standard expression for the number of sets of i things (aside from reordering) from a set of m things is the BinomialCoefficient.

For example see

http://mathworld.wolfram.com/BinomialCoefficient.html

or

http://en.wikipedia.org/wiki/Binomial_coefficient

related to the familiar Pascal's Triangle

My apologies for the (possible) poor formatting of this post... I cannot get the forum to properly format it.... There is a bug in the forum's software....

POSTED BY: David Reiss

How does this tell us that the answer is ten? Also, is there a way to demonstrate how many groups of 2 and 3 there using this formula?

POSTED BY: Trudie Allen

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}]]

enter image description here

POSTED BY: David Reiss
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