3 Sevens of Diamond Odds

I know that a player gets a hand of 21 made of 3 Sevens of Diamonds out of a 6 deck shoe is about 1 in about 250667 deals.
(Now I am not sure if that includes or not when the dealer checks for blackjack, but in my case the dealer has no hole card.)

I would like to know how the above number is affected if there is only 1 player at the table or more (say 7 players playing)?
I think there is less chance of hit if there are more players at the table, opposed to only one player. How to account this aspect?

Also, is the above number calculated from a continuous dealt of 312 cards or takes into account and remove the dealt cards until there are eliminated 4 seven of diamonds out of the shoe and only then starts the counting with all 312 cards again and so on?

Number of other players at the table is irrelevant.

The number is computed as follows:

There are 6 7Ds in the deck, and 312 cards. So there is a 6/312 chance that your first card is a 7D.
If your first card is a 7D, there are 311 cards left in the deck and 5 7Ds left. So there is a 5/311 chance that your next card is a 7D. And, if you hit the first 2, there is a 4/310 chance of hitting the 3rd one.

6/312 * 5/311 * 4/310 = 1 / 250,666 exactly. So the number you gave is off by 1.

Oh, OK thank you Axiom now I am clear about my above questions.