Starting hands against multiple opponents in Texas Hold ‘em

When facing two opponents, for any given starting hand the number of possible combinations of hands the opponents can have is

(50/2)(48/2) = 1,381800

hands. For calculating probabilities we can ignore the distinction between the two opponents holding A♠ J♥ and ♥

♣ and the opponents holding ♥

♣ and A♠ J♥. The number of ways that hands can be distributed between

n

opponents is

n!

(pronounced n factorial). So the number of unique hand combinations

H

against two opponents is

H = (50/2)(48/2) -: 2! = 690,900

and against three opponents is

H = (50/2)(48/2)(46/2) -: 3! = 238,360,500

and against $n$ opponents is

H = ⁿ_k=1Π((50-2k)/2) -: k or alternately H = (50/2n) x (2n-1)!!

where $(2 n - 1)!!$ (!! is the double factorial operator) is the number of ways to distribute $2 n$ cards between $n$ hands of two cards each. The following table shows the number of hand combinations for up to nine opponents.

Opponents	Number of possible hand combinations
1	1,225
2	690,900
3	238,360,500
4	56,372,258,250
5	≈9.7073 × 10¹² (more than 9.7 trillion)
6	≈1.2620 × 10¹⁵ (more than 1.2 quadrillion)
7	≈1.2674 × 10¹⁷ (more than 126 quadrillion)
8	≈9.9804 × 10¹⁸ (almost 10 quintillion)
9	≈6.2211 × 10²⁰ (more than 622 quintillion)

An exhaustive analysis of all of the match ups in Texas Hold ‘em of a player against nine opponents requires evaluating each possible board for each distinct starting hand against each possible combination of hands held by nine opponents, which is

69 x (50/18) x 17!! x (32/5) ~ 2.117 x 10²⁸ (more than 21 octillion.)

If you were able to evaluate one trillion (10¹²) combinations every second, it would take over 670 million years to evaluate all of the hand/board combinations. While it is possible to significantly reduce the total number of combinations by pruning combinations with identical properties, the total number of situations is still well beyond the number that can be evaluated by brute force. For this reason, most software programs compute probabilities and expected values for Hold ‘em poker hands against multiple opponents by simulating the play of thousands or even millions of hands to determine statistical probabilities.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

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