Counting Problems in Physics What are some classic counting problems in physics? I'm trying to think of interesting examples to give in a math class on the matter, and I feel as if physics should have some ones to offer.
 A: I'm not sure if you're talking about probabilities here but if you are then I found some problems online: 
1) In how many ways can 8 people line up for concert tickets?
2) There are 5 women running a race. How many different ways could 1st, 2nd, 3rd place finishers occur? 
3) There are 13 members on a board of directors. If they must form a subcommittee of 6 members, how many different subcommittees are possible?
I apologise if those are too easy, or not what you're looking for.
A: Here is a classic problem connected to the computation of the density of states in statistical mechanics.
A particle of mass $m$ is confined to a three=dimensional cubical box of side $L$. Quantum mechanically, the energy of the particle can take the following values:
$$
E(n_1,n_2,n_3)= \epsilon_0 \left(n_1^2 + n_2^2 + n_3^2\right)\ ,
$$
where $\epsilon_0=\frac{\hbar^2}{8mL^2}$ and $n_1,n_2,n_3=1,2,3,\ldots$ ad inf. Let $\Phi(E)$ denote the number of possible states (values of energy counted with multiplicity) with energy $\leq E$. Determine $\Phi(E)$ for $E=n\epsilon_0$ where $n=1,2,\ldots$.
The problem can be simplified further by reducing the number of dimensions.
A: There are some very good physics problems involved in counting modes. Actually, the cases where you have to actually count the modes are rather advanced, because you come up against the high-frequency cutoff situations and that involves some heavy-duty physics. But listing the modes is a good math problem. 
To do this with a high-school class you'd have to start by understanding the one-dimensional modes of a guitar string. Then the question becomes: can you generalize this to the two-dimensional case of the rectangular rubber membrane? I think the biggest obstacle to doing this with a high-school class would be whether you can make a convincing argument that the natural modes are found by divding the rectangle into sub-rectangles. But if you can get past that, it becomes a good counting problem.
The two interesting generalizations which follow: first, to use a circular membrane. If you've really understood the rectangular membrane, then the general topology of the circular modes should follow logically. You can point out that the exact description of the spacing of the rings is given by what we call "Bessel Functions", but you don't have to do that.
The other interesting generalization is to do the modes on the surface of a sphere. These could represent the way the oceans can heave on a planet which is covered entirely by water. It also, of course, represents the shapes of the s, p, d etc. orbitals of the hydrogen atom. Which I think are actually part of the high school curriculum.
