5. The player in the fifth seat can match up his hand with the players in seats 6, 7, 8, 9, and 10, for a total of five matchups. TOTAL: 9+8+7+6+5=35
6. The player in the sixth seat can match up his hand with the players in seats 7, 8, 9, and 10, for a total of four matchups. TOTAL: 9+8+7+6+5+4=39
7. The player in the seventh seat can match up his hand with the players in seats 8, 9, and 10, for a total of three matchups. TOTAL: 9+8+7+6+5+4+3=42
8. The player in the eighth seat can match up his hand with the players in seats 9 and 10, for a total of two matchups. TOTAL: 9+8+7+6+5+4+3+2=44
9. The player in the ninth seat can match up his hand with the player in the tenth seat for only one matchup. Total: 9+8+7+6+5+4+3+2+1=45
This list means that for every hand dealt, there are forty-five hand matchups. If it takes 12,500 hands to hit a jackpot, then when a jackpot is hit, on average, there will have been a total of 562,500 hand matchups (12,500 x 45 = 562,500). That number applies for a full, ten-handed game.
If the games were all nine-handed, then there would be only thirty-six matchups per hand, and you would have to divide 562,500 by 36 to get the number of hands that it would take to hit the jackpot. That comes out to be 15,625 hands (562,500/36 = 15,625).
With just one player fewer at a table, the odds against hitting the jackpot increase by exactly 25% (12,500 + 25% = 15,625) instead of 10% as you might expect. That's because the number of possible matchups decreases exponentially as each player leaves the game. Using the above math as a foundation, I've created the following table to tell you the odds against hitting a jackpot with various numbers of players in the game.
ODDS AGAINST HITTING A JACKPOT
total divided by # of hands needed
# of players matchups # of matchups to hit jackpot
10 562,500 45 12,500
9 562,500 36 15,625
8 562,500 28 20,089
7 562,500 21 26,785
6 562,500 15 37,500
5 562,500 10 56,250
4 562,500 6 93,750
3 562,500 3 187,500
2 562,500 1 562,500
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