Rule of Division

The rule of division is our next counting principle. Intuitively, it says that if you have n things, but they can be put in groups of size d such that all of the things in a group really count as the same thing, then we have n/d things.

That may be a bit confusing, so let's consider an example.

Example: Bridge pairings

In the card game bridge, there are four players who play in partnerships of two. We might ask how many possible pairings there are for a given set of four players.

One way to count them would be to start by counting all possible sequences of players, with the first two players being a pair and the other two players being a pair.

It turns out that there are 24 such sequences, as we will see in the upcoming section on permutations. But some of these sequences are really just the same thing. For example, the following sequences all give us exactly the same pairings:

  • (1, 2) vs (3, 4)
  • (1, 2) vs (4, 3)
  • (2, 1) vs (3, 4)
  • (2, 1) vs (4, 3)

Since for any given pairing there are four ways to specify it, we can apply the rule of division to conclude that there are 24/4 = 6 possible pairings. You can verify this by examining the enumeration below:

  • (1, 2) vs (3, 4)
  • (1, 3) vs (2, 4)
  • (1, 4) vs (2, 3)
  • (2, 3) vs (1, 4)
  • (2, 4) vs (1, 3)
  • (3, 4) vs (1, 2)

The calculation we just performed is called counting combinations, which are groups of things where the order doesn't matter. We will consider this in more detail after we learn about permutations.