Video Blackjack: odds of getting 4 blackjacks in 7 hands?

I went to the casino recently, and found a machine where you played 7 hands of single deck blackjack, against the dealer. There was a progressive jackpot running that would pay out if you were dealt 4 blackjacks out of your 7 hands. It got me wondering what the odds for that event actually were. I've been running the numbers and I'm not sure if this is right or not, I'd appreciate some feedback.

Here's what I've worked out so far:

((4/52)(16/51)x2) Probability of first blackjack
((3/50)(15/49)x2) Second BJ
((2/48)(14/47)x2) Third BJ
((1/46)(13/45)x2) Fourth BJ in a row

So multiplying all these together gives use the probability of getting 4 black jacks in a ROW.

I'm a little confused as to how to take into account that we have 7 hands, and a dealer.
I thought we'd have to multiply that by 7 choose 4 to indicate only our hands get the blackjack.

Or is it something more elaborate than that?

Am I close? Thanks in advance.

This seems right to me. Take the product of your calculations and multiply by 7 choose 4. You don't need to calculate the rest of the hands because it is guaranteed that they are not blackjack.