Tricki
a repository of mathematical know-how

Counting techniques based on multiplication

If a set can be decomposed into pieces of equal size then the number of elements in it is the multiplication of the number of pieces and the common size of the pieces. This gives a method to compute any of these quantities if the other two are simple to compute: the size of the composite set, the number of pieces, or the common size of the pieces. As trivial as this sounds, it is useful. The first article in the list below describes this idea in detail.

Very often the composite set of interest is a generalized cartesian product. A decomposition of such a set may give pieces that are composite themselves. If the resulting pieces at each decomposition step are the same size, one can still use multiplication and the total count is now obtained as a sequence of multiplication operations. The second article describes situations when this occurs.