Thread: Counting

Ace high hands



Is this correct because someone else said the answer is 502,860.

Hello, dwsmith!

Your answer is off . . . and Someone Else's answer is off, too.

Ace high hands



Is this correct because someone else said the answer is 502,860.

Your answer is correct . . . up to a point.

You said there are 4 choices for the Ace
. . then we select 4 of the other 12 values
. . and there are ways to select those values.

You found the number of hands which contain an Ace and no pairs. . Good!


However, this includes the hands that are Straights or Flushes.


There are Ace-high straights: .
. . There are: . of them.

There are Ace-low straight: .
. . There are: . of them.

Hence, there are: . Straights.


For the Flushes, there are: choices for the Ace.
And choices for the other 4 cards of that suit.

Hence, there are: . Flushes.


It seems that there are: . Straights and Flushes.

And this is what Some Else subtracted.


But he/she forgot that we've overcounted.
There are some hands which are Straights and Flushes.

There are 4 hands that are Ace-high straight flushes.
(They're called Royal Flushes, remember?)

So, the number we subtract is: .


Therefore, there are: . Ace-high hands.

What are your asking about?