In how many ways can I select 13 cards from a standard deck of 52 cards so that 5 of those cards are of the same suit?

Question

My professor says the solution is $47 \choose 8$, but I don't understand how and why. Could you please explain the solution or say whether it is correct or not? Thanks in advance!

Answer 1

Neither can I. Say $A$ is a set of selection of 13 cards with exactly 5 diamonds and similary for $B,C$ and $D$. Then $$|A|= |B|= |C|= |D| = {13\choose 5}\cdot {39\choose 8}$$ and $$|A\cap B | = |A\cap C|=...= |C\cap D| = {13\choose 5}\cdot {13\choose 5} \cdot {26\choose 3}$$ Then by PIE we have $$|A\cup B\cup C\cup D| = 4\cdot {13\choose 5}\cdot {39\choose 8} - {4\choose 2} \cdot {13\choose 5}\cdot {13\choose 5} \cdot {26\choose 3}$$

Answer 2

This is an answer for an IMV reasonable interpretation.

Let $H$ denote the selections that contain exactly $5$ hearts and let there be similar definitions for $D$, $S$ and $C$.

Then to be found is $|H\cup D\cup S\cup C|$.

Applying inclusion/exclusion and symmetry we find that this equals:$$4|H|-6|H\cap D|=4\binom{13}5\binom{39}{8}-6\binom{13}5^2\binom{26}3$$

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This gives you the number of selections characterized by the fact that at least one of the suites is represented by exactly $5$ cards.

Answer 3

It is plausible that either your professor misunderstood one detail of the question, or you misheard the question, and thus accidentally omitted a detail. As such, I present what might have been "what was your professor actually giving an answer to":

In the context of a deck of cards, $\binom{47}{8}$ would most naturally be interpreted as the number of ways you could choose 8 cards from the deck after removing 5 specific cards. It is easy to think that the 5 specific cards are part of the "selected" set, and thus the total set is 13, made up of the 5 specified cards plus 8 other cards.

With this, it is plausible that your professor was thinking of "five cards of the same suit" as five specific cards, and it doesn't matter if some of the other cards are of the same suit as those five.

In particular, it would be easy for your professor to inadvertently think it was "how many ways can I select 13 cards from a standard deck of 52 cards so that the 5 'face' cards of one suit are all present". Alternatively, it would be easy for you to miss those specific details.

It is difficult to see other ways to obtain a similar answer. It's possible, however, that your professor mistakenly considered the question to be comparable to similar-sounding questions like how you could seat $x$ people so that a group of $y$ friends are seated together.