A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, the superdeck has 52 · 10 = 520 cards, with 10 copies of each card. How many different 10-card hands can be dealt from the superdeck? The order of the cards does not matter, nor does it matter which of the original 10 decks the cards came from. Express your answer as a binomial coefficient.

This question is already answered in the link below:

Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)

But I still have questions:

1.What does a 10-card hand mean? Does it mean all cards of the same kind?

2.What if the question had 11 decks instead of 10. Would $$\binom {52+11-1}{10}$$ be correct?

3.How are there 52+11-1 slots instead of 52*10=520 slots for 10 deck of cards?


"all cards of the same kind?" No. A hand of cards is just a subset of the cards. Here, since we have duplicate cards we need to be a bit more careful with how we phrase it... it is a submultiset of the cards. Order within the hand does not matter. Two hands are considered the "same" if what cards in what quantities are the same in one hand compared to another. For example, a hand that contains five $A\spadesuit$ four $K\heartsuit$ and one $K\clubsuit$ is considered the same as any other hand which contains those same type of cards in the same quantities, regardless order and what deck. Here it is assumed that which deck a card comes from is irrelevant. If every deck of cards had different art styles and a different cardback and could be distinguished it would have been a very different problem, one with an answer of $\binom{520}{10}$

"What if the question had 11 decks" No, the answer will still be $\binom{10+52-1}{52-1}=\binom{10+52-1}{10}$, we did not need an $11$ there. The $10$ that appeared in the formula here happened to be our handsize not the number of decks. That the number of decks was ten is only relevant because it is greater than our number of cards in hand, allowing us to treat this as though it were pulling with replacement. If it were five decks instead, we would need to be more careful, likely needing to combine stars-and-bars with inclusion-exclusion.

"how are there $52+10-1$ slots instead of $52\cdot 10=520$ slots for ten decks of cards?" I don't know what you mean by slots here, but this is again a standard application of stars-and-bars. We have different categories. We want some collection of some number of those categories, repetition allowed order not relevant. In order to split the categories up, we only need a barrier between the categories.

Consider this small example of having five identical cookies and we ask how we can split them up between me and you. I say, we can draw a line. All cookies on the left side of the line are mine and all cookies on the right side of the line are yours. We have the possibilities: $\mid \circ \circ \circ \circ\circ$ where I get none and you get five, we have $\circ \mid \circ \circ \circ \circ$ where I get one and you get four... on up to $\circ \circ \circ \circ \circ \mid$ where I get all five and you get none. I only needed one barrier to separate these into two piles, not two barriers. Similarly, if I wanted to split it into $k$ piles I only need $k-1$ barriers.

The essence of stars-and-bars is again to distribute $n$ identical balls (cards in our hand) into $k$ distinct bins (cardtypes) where order in which balls appear in bins doesn't matter this is like picking some arrangement of $n$ stars and $k-1$ bars which can be done in $\binom{n+k-1}{k-1}$ ways, equivalently $\binom{n+k-1}{n}$ ways.

  • $\begingroup$ So...I treat the 10 card-hand as one cookie each and the identical card sets (52) as the barriers and then apply stars and bars? $\endgroup$
    – Orpheus
    Oct 17 '20 at 13:09
  • $\begingroup$ You need $51$ barriers to separate into $52$ categories, each category corresponding to a different rank-suit cardtype, the ten cards from the ten-card hand as one cookie each, not the hand itself as a single object. $\endgroup$
    – JMoravitz
    Oct 17 '20 at 13:25

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