There are 8 types of Made Hands: pair, two pair, three of a kind, straight, flush, full house, quads and straight flush.
Given that 8 players see the river card, is it possible that each has a made hand with all 8 types represented?
Spoiler:
NO
In order for a player to have quads (or a FH), the board has to have a pair. In that case, one cannot have a set or trips .
Example:
Board: 2c 2h 3c 4c 5c
For a player to have a set or trips, he cannot have a 2 for then quads are not possible. If he has a pair matching one of the other 3 board ranks, he has a full house. All other made hand types are possible with this board.
I sometimes wonder if I spend too much time thinking of inconsequential things.
In order to have a full-house, the board must be paired. In order to have one pair, the board must have no more than one pair. Two exact cards are required to make quads on a single paired board, and in order to make trips one of those cards would be needed. This is not possible in Hold'em.
Spoiler:
However, this applies to Hold'em. In Omaha, in order to have a full-house, the board must be paired, however, the board can be double-paired and still allow a player to hold one pair. There turns out to be enough cards available to make each type of hand. Here is an example:
Board:
AKQAK
Hands in descending order:
JT22
AA22
QQ33
4433
JT44
KJ55
6655
9876
K
K
2