A Combinatorics Problem

There is a kingdom called Qidan that recently conquered three cities A, B and C. King Temutai wants to construct roads to connect the Capital to these three cities. At present, no two of these four places have road(s) connecting them.

The king has decided to construct a total of 630 roads, each of which fall into exactly one of the following three categories:

  • From the Capital to city A;
  • From city A to city B and
  • From city B to city C.

The King wishes to maximize the number of different ways from the Capital to city C, that passes through A and B exactly once. What is the maximum number of ways?

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