because the burned cards and the other hands are unknown to you, they do not count and therefore you do not need to deal with them in your odds dealing. As far as those cards are concerned, they are just part of the deck.

I would suggest we start with the assumption that you do not know your own cards either and just go with the basic formula:

(4*3*2*1)/(52*51*50*49)

You have four options on the first draw out of 52 cards, 3 remaining sixes out of 51 cards, and so on. The value of this is:

24/6497400 or one out of 270725.

NOW: If we want to be even more specific, we could say something along the lines of "the first nine cards cannot be sixes, the next three must be sixes, the next card not a six, and the last card a six". This would include the hands being dealt, the burn cards and the exact order of the sixes. it would look something more like this:

(48*47*46*45*44*43*42*41*40*4*3*2*39*1)/(52*51*50*49*48*47*46*45*44*43*42*41*40*39)

Simplified, we can cancel out some of the numbers on the top and the bottom: 48 through 39 get removed because they're on the top and the bottom. We are left with:

(4*3*2*1)/(52*51*50*49)

Look familiar?