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.