In Cribbage, a hand scores as follows:

2 points for each set of cards that totals 15 (face cards count 10, aces count 1)

2 points for each pair (this means 3-of-a-kind is worth 6 points and 4-of-a-kind is worth 12 points)

n points for each maximal straight containing n cards

(i.e. a four card straight does not also count as two three card straights)

n points for each maximal flush containing n cards

(i.e. a four card flush does not also count as four three card flushes)

1 point for the jack of trumps

It's easy to show that the best five card hand is J5555, worth 29 points, and, although impossible in an actual game, the best six card hand would be 445566, worth 46 points.

If the entire deck of 52 cards was considered to be a single cribbage hand, what would be its value?

For each category Cribbage requires the largest of a set held in a hand to be scored; there can be no splitting, so no "double-dipping".
[Note dot points 3 and 4, the concept is somewhat noted in point 2 as well].

We score on Pairs, Straights, Flushes and Fifteen counts.

Pairs - These are contained in 13 sets of 4 of a kind, each worth 12 points, thus  156 points for pairs.

Flushes - There are 4 of each suit of 13 cards
        and thus 4 * 13 = 52 points

Fifteens - Only the cards from 5 though K are used here.
    K, Q, J and 10 all have a face value of 10.

There are 4 'value' cards that you can choose for your 10, but there are 4 of each of those cards.  For each of these ways you have any of the 5's available to make your 15;
        ie 4^5 ways * 2 points = 2048 points

Then there are the combinations of 9 with 6 and 8 with 7.  These represent 16 ways each.  With their points value that is 64 points.

Now this could be easily missed.  Four 5's have been counted in the pairs, but it is legitimate to count them here again as 15's, we are not "double dipping" - 12 points.

Straights - We have 4 cards available for each position in our straight (we don't have to worry about suits).  Thus we have 4^13 ways to create our straights. With the 13 points score this then is 4^13 * 13 = 872415232 points

Total  872415232 + 12 + 64 + 2048 + 52 + 156 =
                872417564 points all up.

I note that what I offer (other than the 'jack of trumps') differs with Charlie's computer solution in the matter of the 15 count.

I understand the variables of the program, and in his explanation at the end he seems tohave a fixation on suits which isn't necessary.

Edited on June 7, 2006, 10:55 pm

Posted by brianjn on 2006-06-07 21:09:39