# Social dynamics

Our little group here in the woods consists of four people. As I’ve been watching the various conversations develop, I’ve been observing how the social dynamics change, depending upon who is or isn’t present in any particular discussion.

Counting all the possibilities, I see that there are eleven possible conversational groupings among four people: (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) (1,2,3) (1,2,4) (1,3,4) (2,3,4) (1,2,3,4).

Between any two people there is only one possible set of groupings, and between any three people there are four: (1,2) (1,3) (2,3) (1,2,3).

There turns out to be a simple formula based on the number of people, which is how I know that for five people there are 26 possible conversational groups, for six people there are 57, and for seven people there are 120. **

I will leave it up to you to figure out what the formula is.

** My original post contained typos for some of these numbers, which Sharon and Logan kindly corrected.

## 6 thoughts on “Social dynamics”

2^n (subsets) -n (no talking to yourself) -1 (if a conversation is held by no one does it really take place?)

2. sharon says:

I came about it from a less elegant approach than Adam, but it gives the same values:

n n
SUM (n choose k) = SUM (n!/k!(n-k)!
k=2 k=2

(that is, summing over all sizes of groups of at least 2, the number of ways to form a group of that size given n people).

However, both my formula and Adam’s give 26 for n = 5, and 57 for n = 6. Ken?

3. sharon says:

Yuck, that didn’t format well. The second formula is

n
SUM (n!/k!(n-k!))
k=2