*How many ways can k indistinguishable balls be placed in n distinguishable boxes?*

To answer this, you take a 'star' for each of the k balls, and n - 1 'bars' denoting the separators between the n boxes. Then each arrangement of the balls corresponds to a permutation of these k + n - 1 objects. For instance, when k = 6, n = 4, one of the arrangements might be written (3, 0, 2, 1); that is, 3 balls in the first box, etc. This arrangement corresponds to the string

*** * * | | * * | ***

of 6 stars and 3 bars. But now that we have only two kinds of objects, it is easy to count the number of arrangements: there is one arrangement for every choice of *which k out of the k + n - 1 slots receive a star*. Thus the answer is the binomial coefficient C(k + n - 1, k) = C(k + n - 1, n - 1).

Among other things, this number counts the multi-indices of dimension n and weight k, which is the dimension of the space Sym^{k}(**R**^{n}) of symmetric tensors of type (k, 0) or (0, k) on **R**^{n}. This space comes up when you formulate multivariable calculus in a general context, as the kth derivative of a function **R**^{n} → **R** is a symmetric (k, 0) tensor.

A good place to learn more about this kind of problem is *Concrete mathematics* by Ronald Graham, Donald Knuth and Oren Patashnik. Richard Stanley's two-volume masterwork *Enumerative combinatorics* is a more advanced reference; the stars-and-bars problem is one component of what he calls the Twelvefold Way of basic counting.