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 andno 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'veovercounted.

There are some hands which are StraightsFlushes.and

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.