I am investigating a game called Rummy Royale. If I set the number of players to $3$, then I can split the deck of $52$ cards into $4$ hands of $13$ cards each.
When choosing $13$ cards from a deck of $52$ cards, what is the probability of choosing precisely the cards with the numbers $6,7,8$ of the same suit? For example a $6,7,8$ of heart or a $6,7,8$ of diamonds, or a $6,7,8$ of spades, or a $6,7,8$ of clubs. All three cards must be the numbers of $6,7,8$ and only of a single suit. These must be among the hand of $13$ cards.
My first thoughts on this question that I only have $4$ possibilities of this happening for the entire deck of $52$ cards. So I thought $(_4 C_1)(_{13}C _{3})$ will account for the possible number of decks that will include cards of $6,7,8$ of all the same suit out of $13$ cards in the hand.
And $(_{52} C_{13})$ gives me all the possible combinations of 13 cards from 52 cards. So this probability that I got would be: $\frac{(_4 C_1)(_{13}C _{3})}{(_{52} C_{13})}\approx 5*10^{-7}$. However I feel something is off. I haven't done probability in over a decade.