Calculating the Probability of a Royal Flush in Texas Holdem with 9 Players

[Zythyr]

How would I calculate the probability of getting a royal flush in a game of Texas Holdem with 9 players sitting on a table playing not against each other (without betting), but with the goal of producing a royal flush on the table. It doesn't matter which player gets the royal flush. The goal of the whole table is accomplished when a royal flush is attained.

 

[Fredrik]

I think you will have to consider a lot of different cases separately. Only one of them is really easy, so I'll just do that one. The probability that the board will be a royal when we have no information about the hole cards is

$$\frac{20}{52}\cdot\frac{4}{51}\cdot\frac{3}{50}\cdot\frac{2}{49}\cdot\frac{1}{48}=\frac{1}{649740}$$