[BruceZ]

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I'd say that if the board in HE is 'royal friendly' that increases the chance that all 7 will get a royal. And if the board is 'royal unfriendly' it decreases the chance of all players. I'd think (without doing the math) that that exactly outweighs each other, so you can do the math for 7 independent hands and it gives the same results.

But that's just an assumption I have no facts or math to base on.

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In order for you to multiply by 7, you need the 7 hands to be mutually exclusive, not independent. Mutually exclusive means that only 1 player can have a royal. This is not quite true because when there is a royal on the board, all 7 players have a royal. By multiplying by 7, you are counting the cases where there is a royal on the board 7 times. You must subtract off 6 of these as I did in my solution so that you only count these once. These are the only cases where more than 1 player can have a royal.