From this link
The number of distinct 5-card poker hands that are possible from 7 cards is 4,824.
Does anyone have the break down of the 4,824?
A list of the 4824?
Better yet a program to generate the 4824?
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The hands that can't happen with seven cards are exactly the five card hands that improve no matter which two cards are added to them.
This basically means hands with low kickers.
A high card hand (think of very card as a kicker) like 7 5 4 3 2 is impossible because any cards added to it will improve it.
Some classes are easy to show that there are always two cards which can be added without improving it, and thus all of those hands are possible in 7 card poker:
All of those hands can occur in 7 card poker. The hands below that all have some cases where there is no choice but to improve on the 5 card hand by adding two more. For example: Three of a Kind. If the kickers are both high enough there is a way to avoid improving the hand. Hands with at least one low kicker like AAAK2 or 44432 can only be improved though and thus can't occur in 7 card poker. AAAK3 also can't avoid being improved because we need to add two cards, and once we've added a 2 there is no second card that can be added. In most cases if there is a 2 or a 3 as a kicker the hand can't occur in 7 card poker, but there are a few cases where there doesn't need to be a 2 or a 3 either, like AAA45, where adding a 2 and 3 would form a straight.
The fourth column in the link that Lee Daniel Crocker posted is the number of ways for that hand to occur in 7-card poker. Anywhere that column has a non-zero is a hand that is possible. All 1609 hands that are straights or better are possible. Of the remaining 5853 hands 3215 are possible.