# Number of possible Straights in a deck of 52 cards

1. Oct 22, 2012

### audifanatic51

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

2. Oct 23, 2012

### FeynmanIsCool

"There are 14 effective ranks of cards for a straight (ace can be high or low). The straight can start on any one 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.

3. Oct 23, 2012

### haruspex

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.

4. Oct 23, 2012

### audifanatic51

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?

5. Oct 23, 2012

### haruspex

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.