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There are 20 unique 5-card hands.
Solution:
First break apart the player card combinations.
You can either use player card 1, player card 2, or both player cards 1 and 2 with the hand. Therefore, you are taking 5 cards, 3 at a time, when playing with both player cards, and 5 cards, 4 at a time, when playing with 1 player card.
Mathematically, this can be expressed using the Binomial coefficient
2 x ( 5 choose 4 ) + ( 5 choose 3 ) = Number of possible 5-card hands.
5 choose 4 = 5 : This represents the combinations between 1 player card and 5 table cards, 4 at a time. (http://www.wolframalpha.com/input/?i=5+choose+4)
5 choose 3 = 10 : This represents the combinations between 2 player cards, and 5 table cards, 3 at a time. (http://www.wolframalpha.com/input/?i=5+choose+3)
Thus, 2 x ( 5 ) + ( 10 ) = 20 .
Further Information:
All possible hand combinations are listed below.
A and B are the two player cards.
0 through 5 are the five table cards.
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A 1 2 3 4
A 2 3 4 5
A 3 4 5 1
A 4 5 1 2
A 5 1 2 3
B 1 2 3 4
B 2 3 4 5
B 3 4 5 1
B 4 5 1 2
B 5 1 2 3
A B 2 3 4
A B 3 4 5
A B 1 2 3
A B 4 5 1
A B 5 1 2
A B 4 5 2
A B 4 1 2
A B 5 1 3
A B 5 2 3
A B 1 3 5
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What is the starting hands poker chart and how can it help improve my game?
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