# wording question regarding a 13 cards game $\binom{52}{13}$ vs $\binom{52}{1}$ vs $\binom{52-4}{1}$

I have a wording question regarding the following question:

"Four players, named A, B, C, and D, are playing a card game. A standard, well-shuffled deck of cards is dealt to the players (so each player receives a 13-card hand).

How many possibilities are there for the hand that player $$A$$ will get? (Within a hand, the order in which cards were received doesn't matter.)"

$$\require{cancel}$$ From my understanding, the possibilities for player $$A$$'s hand is $$\binom{52}{13}$$ if the cards are distributed in groups of 13s. However, if the cards are distributed one after another to each of the four players, the possibilities will be $$\cancel{ \binom{52}{1}+\binom{52-4}{1}+\binom{52-8}{1}\cdots\binom{52-48}{1} }$$ correct?

Sorry I'm not familiar with how most card games are distributed among players, hence, the question. And does "within a hand" simply means all the cards in hand during the game?

$$\binom{n}{k} = \binom{52}{13} = \frac{\prod_{i=0}^{n-1} \binom{52-i}{1}}{13!}$$
• Since the player does not see the cards the other players get, you may want ${52 \choose 1} \times {52-1 \choose 2} \times \cdots \times {52-12 \choose 1}$, which is $13!$ times the first result as here order did matter. Apr 25, 2022 at 12:10
• With regards to $(52-1)$ options versus $(52-4)$ options for the "second" card that player $A$ gets... while yes player $A$ can not get as their second card any of the cards that players B,C,D got for their first... given a first card for $A$ and if we don't keep track of what those were that the other players got it still remains the case that any of the $51$ cards are equally likely to be player A's second card. See this. Apr 25, 2022 at 12:20
• @JMoravitz so the answer to (52-1) vs (52-4) depends on whether the cards are in player $A$'s hand. Hence, cards that are/were distributed to players B,C,D are considered "still" in the deck when calculating the possibilities. Therefore, the correct way to to view (n-k); is k = number of cards player $A$ has in hand, not the total number of cards distributed. Apr 25, 2022 at 13:05