Question

Suppose you are playing a game of five-card poker where you are dealt a hand of 5 cards from a standard deck of 52 playing cards. Each card has one of 13 values, 2,3,…,10,J,Q,K,A2,3,…,10,J,Q,K,A, and one of 4 suits, ♡,♢,♠,♣♡,♢,♠,♣

A hand called a royal flush consists of cards of the values 10,J,Q,K,A10,J,Q,K,A, all from the same suit. For instance, a royal flush with hearts would be

10♡,J♡,Q♡,K♡,A♡10♡,J♡,Q♡,K♡,A♡.

Now, actually getting a royal flush in poker is incredibly rare (in fact, it should be easy to see see that there are only 4 ways that you can draw a royal flush). However, to people that play poker occasionally, it seems like almost getting a royal flush is all too common. It's this scenario that we'll be interested in in this problem. We'll answer the question: "How many ways are there to pick a 5-card poker hand that is one card away from being a royal flush?"

For example, if you have 10♣,J♣,Q♣,K♣,A♡10♣,J♣,Q♣,K♣,A♡, the hand is one card away from being a royal flush.

Similarly, if you have 10♢,J♢,4♣,K♢,A♢10♢,J♢,4♣,K♢,A♢, the hand is also one card away from being a royal flush.  

So we want to figure out how many such hands there are. Don't worry - we're going to walk through this together! We can break this task of creating an "almost-royal-flush" hand into a few steps:

Figure out which 4 of the 5 royal flush card values we've gotten right

Figure out what is the suit of the 4 royal flush cards that we have

Figure out what the last card is - that card that ruined your royal flush!

Figure out how to combine Steps 1-3 to get our answer

Part A. First, completely ignoring suit, how many ways are there to pick which 4 of the 5 royal flush card values are in your hand?



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Part B. Alternatively, how many ways there are to pick which 1 of the 5 royal flush card values is not in your hand?



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Part C. Look at your answers to Parts A and B. Notice anything? Do not proceed until you notice something interesting.

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Part D. Okay, so we've dealt with the number of ways to get 4 out of the 5 royal flush card values. But this is an almost-royal-flush we're talking about, so at least four of these cards should be the same suit. How many ways are there to decide which suit an almost-royal-flush will be?

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Part E. Great! Now we have the almost-royal-flush cards' values and suits. So those 4 cards are completely determined, but we still need to decide what the other card should be.

After we selected the first 4 cards, there are 48 cards left in the deck. One of these 48 cards, however, will leave us with an actual royal flush if we pick it. How many ways are there to choose our fifth, and final, card such that we do not end up with a royal flush?



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Part F.  The procedure of creating a poker hand that is one card away from being a royal flush consists of 3 steps, all of which must be completed in order to accomplish the task. Use the appropriate rule (product rule or sum rule) to calculate the total number of 5-card poker hands that are one card away from being a royal flush, and your answers from the previous parts.



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Part G.  How many 7-card poker hands are one card away from containing a 5-card royal flush?  Hint: the first two steps of this procedure are the same as the 5-card problem; it is the last part - picking the card(s) that ruined your royal flush - that needs to be modified.

Answer 1

a) The number of ways to pick which 4 of the 5 royal flush card values are in your hand is (ignoring suit).

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d) There are 4 suits. The above number should be multplied by 4.

e) The number of ways to choose our fifth, and final, card such that we do not end up with a royal flush

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f) Use product rule to calculate the total number of 5-card poker hands that are one card away from being a royal flush, and your answers from the previous parts.

g) The total number of 7-card poker hands that are one card away from being a royal flush, and your answers from the previous parts. is found analogously,