Day 13 Play-by-Play

There were two key ideas:

• Configurations of Facebook, ignoring identity -- and we discover that there are 11 of them, as seen in this figure:
  

  We talked again about duality between the configurations, and how to get from one to another. The key idea is if there is a connection (edge) in one graph, you break it; if there's no connection, you make it; and that produces the dual.

  So the completely isolated graph is the dual of the complete graph, etc.

• The second part is that was that Pascal's Triangle contains information about "how many ways?"

  This is important for gambling -- lots of games. How many ways of rolling doubles in dice? How many ways of getting a heart flush in cards?.

  In this case, we're looking for the number of Facebooks if we have named individuals (it's always a power of 2!). But that's no surprise, since all the different configurations are given in a row of Pascal's triangle.

• The upshot is that for a Facebook of $n$ individuals, you will go to the row of Pascal's triangle $\frac{n(n-1)}{2}$ (starting from the zeroth row at the top of the pyramid). Then you can read off exactly how many Facebooks you will have of each type (zero friendships, one friendship, etc.):
  

• I illustrate the procedure with an illustration of the different ways to invite five friends to a dinner party. Pascal provides the answer. Again, these sorts of calculations are very important.

  For example, to find out how many poker hands you can make from a standard 52 card hand, you go to the row of Pascal's triangle that starts "1 52 ..." and you pick off the sixth entry in ("How many ways to choose five cards from 52?"): that number, if you were to write Pascal's triangle out to that row(!), would be \[ 2598960 \textrm{ hands} \] By the way, that works out to 52*51*50*49*48 divided by 5*4*3*2*1. And from that, you might be able to guess how every number in Pascal's triangle is computed! (Remember that the triangle is symmetric).