Lin's Game
Alin, Blin, Clin, Dlin, Elin, and Lin took 1, 2, ..., 6 dollars respectively and gathered in the evening to play a card game. The game is designed so that during the evening the total amount of money at the gaming table does not change and at any time all the players have a non-negative integer amount of money. Player $ X $ is upset if he/she loses more than a dollar at the end of the game. It turned out that at the end of the game evening, none of them were upset, and at the same time everyone has a different amount of money. How many different finishing rankings are there in which this is possible?
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