How many different hands of poker can you make with 7 cards?

Answer 1

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