I have an addiction of sorts - I can't keep from playing Freecell. (Most "Windows" users have access to this game from their "Start" menu.) There was once a theory that every possible deal is winnable in this game (this has apparently been disproven).
How many essentially different deals are there in Freecell?
Freecell setup: deal a standard 52 card deck out into eight columns: four with seven cards and four with six. Two deals with only column order changed (i.e., that can be made identical by only switching the locations of particular columns) are not considered different in this context.
Pick 28 cards for the four 7-card columns: this can be done in 52!/(52-28)! ways. The resulting columns can be ordered in 4! ways, so there are only 52!/(24!4!) different ways to set up the 7-card columns.
The remaining 24 cards can be set in 24! ways. As with the 7-card columns, there are 4! ways to order the columns, so there are only 24!/4! ways to set up the remaining 6-card columns.
Multiplying both numbers and simplifying gets the answer: 52!/(4!4!).
Solution
