In Red Dead Redemption 2 there is a challenge to win 3 hands of blackjack while hitting 3 or more times. I'm trying to calculate the probability of this happening on any given hand.
So far using a helpful script from here: https://stackoverflow.com/a/39178939/9044877 I've been able to find the number of possible winning hands at each number of cards to be this:
11: 4
10: 432
9: 5164
8: 22320
7: 48356
6: 58480
5: 38492
4: 12704
3: 2036
2: 64
I'm pretty sure the next step is figuring out the total number of possible combinations for each number of cards similar to how it is done here: Odds of getting the largest possible hand in blackjack and then calculate the odds for each number of cards, then add them up.
The issue with this (I think) is that it doesn't take into account that it doesn't take into account that you would be bust on certain scenarios. To find the number of ways to organize 52 cards into 11 spots you use $\binom{52}{11}$, but this disregards that if you got say 3 kings on your first 3 cards you wouldn't be able to go further.
Is there an intuitive way to solve this that I'm missing?