Number of possible Straights in a deck of 52 cards

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Discussion Overview

The discussion revolves around calculating the number of possible straights in a standard deck of 52 playing cards, specifically focusing on a 5-card hand. Participants explore different approaches to counting the combinations and clarify the conditions under which certain cards can be considered part of a straight.

Discussion Character

  • Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant presents a calculation method involving choosing one card from the deck and then selecting the next four cards based on suit, but questions the validity of this approach.
  • Another participant suggests that there are 14 effective ranks for a straight and identifies 10 base straight sequences, leading to a total of 10 * 4^5 = 10240 possible straights, while noting that 40 of those are straight flushes.
  • There is a discussion about the interpretation of the "first" card in a straight, with some participants arguing that it can be at various positions in the sequence, while others emphasize the need to specify its position to avoid ambiguity.
  • Clarifications are made regarding the number of cards that can serve as the starting point for a straight, with a focus on the limitation of only 40 cards being valid for this purpose.

Areas of Agreement / Disagreement

Participants express differing views on the correct method for calculating the number of straights, and there is no consensus on the validity of the initial approach presented. The discussion remains unresolved regarding the correct counting method.

Contextual Notes

Some assumptions about the definition of a straight and the role of suits in the calculation are not fully explored, leading to potential gaps in understanding the counting process.

audifanatic51
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Hi,

I'm a bit thick-headed I guess and I cannot seem to figure out why my answer to this question is wrong.

I need to figure out the number of ways to obtain a straight in a deck of 52 cards (I do not need to ignore straight-flushes or royal flushes).

Note: a hand of 5 cards is used

My answer was to choose 1 card from a deck of 52 cards and then restrict the next 4 cards to one of 4 possible cards (one for each suit).

So, in short:

52C1 * (4C1)^4

which is wrong. Can anybody explain to me why I am wrong. I'm a-Ok with calc 1-4 and graph theory, but counting theories have never been by favorite, even as a kid. Thanks for the help
 
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"There are 14 effective ranks of cards for a straight (ace can be high or low). The straight can start on anyone of A,K,J,Q,T,9,8,7,6,5 and go down. That makes 10 base straight sequences. Each card in the sequence can be any of the four suits. So the total number of straights is 10 * 4 ^5 = 10240.

40 of those are a straight flush." -Wiki answers.
 
audifanatic51 said:
My answer was to choose 1 card from a deck of 52 cards and then restrict the next 4 cards to one of 4 possible cards (one for each suit).

So, in short:
52C1 * (4C1)^4
You appear to be taking that first card as the top (or maybe the bottom, but not either/or) of the run. As FeynmanIsCool points out, only 40 of the cards can serve as that.
 
haruspex said:
You appear to be taking that first card as the top (or maybe the bottom, but not either/or) of the run. As FeynmanIsCool points out, only 40 of the cards can serve as that.

Ok, you you're basically saying that there really isn't a "first" card since that "first card" can be at the top, end, or somewhere in the middle since order doesn't matter?
 
audifanatic51 said:
Ok, you you're basically saying that there really isn't a "first" card since that "first card" can be at the top, end, or somewhere in the middle since order doesn't matter?
No, I'm saying that if you are taking the first card to define the set of four others then you need to specify where in the sequence that first card comes. You can make it the top, the bottom, or anywhere in between provided you are consistent. But once you have fixed that, there are only forty cards it can be.
 

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