come up heads or tails, . If we toss it twice, it can come up HH, HT, TH, or TT, four ways, and . To ask the specific question about, say, four coins coming up all heads, we calculate . Two heads and two tails would be . We are solving the equation: An important point is that as n gets larger, our distribution gets more and more peaked, and it is more and more reasonable to think of a most likely state. However, the notion of most likely state comes not from this thinking; it comes from statistical mechanics, a field well known to Wilson and not so well known to transportation planners. The result from statistical mechanics is that a descending series is most likely. Think about the way the energy from lights in the classroom is affecting the air in the classroom. If the effect resulted in an ascending series, many of the atoms and molecules would be affected a lot and a few would be affected a little. The descending series would have many affected not at all or not much and only a few affected very much. We could take a given level of energy and compute excitation levels in ascending and descending series. Using the formula above, we would compute the ways particular series could occur, and we would conclude that descending series dominate. That is more-or-less Boltzmann's Law, That is, the particles at any particular excitation level j will be a negative exponential function of the particles in the ground state, , the excitation level, , and a parameter , which is a function of the (average) energy available to the particles in the system. The two paragraphs above have to do with ensemble methods of calculation developed by Gibbs, a topic well beyond the reach of these notes. Returning to the O-D matrix, note that we have not used as much information as we would have from an O and D survey and from our earlier work on trip generation. For the same travel pattern in the O-D matrix used before, we would have row and column totals, i.e.: Consider the way the four folks might travel, 4!/(2!1!1!) = 12; consider three folks, 3!/(0!2!1!) = 3. All travel can be combined in 12×3 = 36 ways. The possible configuration of trips is, thus, seen to be much constrained by the column and