Texas Hold 'em is a form of poker where each player has 2 "pocket" cards, which they combine with 3 community cards from a pool of 5 to form their hand, which are then scored under poker ranks.
Poker cards are scored on the following hands, from best to worst.
- Straight flush (5 consecutive cards of the same suit)
- Four-of-a-kind (4 cards of the same rank, and one other card)
- Full House (3 cards of the same rank, and 2 cards of another rank)
- Flush (5 cards of the same suit, not consecutive)
- Straight (5 consecutive cards, not suited)
- Three of a Kind (3 cards of the same rank, 2 cards of other different ranks)
- Two pair (2 cards of the same rank, 2 cards of another rank, and 1 card of yet another rank)
- One pair (2 cards of the same rank, 3 cards of different other ranks)
- High Card (5 cards of different ranks, neither suited nor consecutive)
For the purposes of this puzzle:
- Players are required to select the highest possible hand. If they have 2 spades as their pocket and there are 3 spades in the community cards, they cannot form any hand worse than a flush.
- All hands of the same type are equivalent. E.g. all straight flushes are treated as the same hand, even ace-high royal flushes.
Is it possible to form a set of community cards and 9 sets of pocket cards so that all 9 kinds of scoring hands are formed? If not, what is the maximum number of different kinds of hands that can be formed?
