Poker Rank Multi-Deck

From WoO's explanation on poker math, the combination for no-rank hand in a single deck 5-card poker is expressed as follow:

(COMBIN(13,5) - 10) * (COMBIN(4,1)^5 - 4)

WoO site has a table showing numerical combinations for single deck to 8 decks, but I want to know the equation for two decks. What is the equation for two decks or any multi decks?

Quote: 777

From WoO's explanation on poker math, the combination for no-rank hand in a single deck 5-card poker is expressed as follow:

(COMBIN(13,5) - 10) * (COMBIN(4,1)^5 - 4)

WoO site has a table showing numerical combinations for single deck to 8 decks, but I want to know the equation for two decks. What is the equation for two decks or any multi decks?

(COMBIN(13,5) - 10) * (COMBIN(4*d,1)^5 - 4) where d is the number or decks.

Quote: miplet

(COMBIN(13,5) - 10) * (COMBIN(4*d,1)^5 - 4) where d is the number or decks.

Thanks Miplet.

Your multi-deck equation yields a no-hand combination of 41839628 for two decks, but a table from WoO sites shows a combination of 41681280.

Link to WoO's Poker Combinations for 1 to 8 Decks Table
https://wizardofodds.com/games/poker/

Quote: 777

Thanks Miplet.

Your multi-deck equation yields a no-hand combination of 41839628 for two decks, but a table from WoO sites shows a combination of 41681280.

Link to WoO's Poker Combinations for 1 to 8 Decks Table
https://wizardofodds.com/games/poker/

Ooos. Try
=(COMBIN(13,5) - 10) * (COMBIN(4*d,1)^5 - 4*d^5)

Quote: miplet

Ooos. Try
=(COMBIN(13,5) - 10) * (COMBIN(4*d,1)^5 - 4*d^5)

Followed your example for no-rank equation, I then developed multi-deck equations for all other ranks, but the flush, three of a kind, 2 pairs and one pair ranks’ results based on my equations shown below do not match the results from WoO’s site. I'm trying to figure out my mistakes. What did I do wrong?

Flush: (this equation appears correct for 1 & 2 decks, but incorrect for 3 decks and higher)
COMBIN(4,1)*(COMBIN(13*d,5) - COMBIN(10,1)*COMBIN(1*d,1)^5)

Three of a kind: (appears correct for 1 & 2 decks, but incorrect for 3 decks and higher)
COMBIN(13,1)*COMBIN(4,3)*COMBIN(12,2)*COMBIN(4,1)^2

Two pairs: (appears incorrect for 2 decks and higher)
COMBIN(13,2)*COMBIN(4,2)^2*COMBIN(11,1)*COMBIN(4,1)

One pair: (appears in correct for 2 decks decks and higher)
COMBIN(13,1)*COMBIN(4,2)*COMBIN(12,3)*COMBIN(4,1)^3

Lets do the easy one first.
Flush

Pick suit combin(4,1)
Pick 5 cards from all of that suit combin(13*d,5)
But the 5 cards can't make a royal or straight flush combin(10,1)*d^5
Or 5 of a kind combin(13,1)*combin(d,5)
Or quads combin(13,1)*combin(d,4)*combin(12,1)*d
Or fullhouse combin(13,1)*combin(d,3)*combin(12,1)*combin(d,2)

Here is the final formula. It works on 5 or more decks. Delete the 5 of a kind portion for less than 5 decks, quads for less than 4 and fullhouse for less than 3.


=COMBIN(4,1)*(COMBIN(13*D1,5)-COMBIN(10,1)*D1^5-COMBIN(13,1)*COMBIN(D1,5)-COMBIN(13,1)*COMBIN(D1,4)*COMBIN(12,1)*D1-COMBIN(13,1)*COMBIN(D1,3)*COMBIN(12,1)*COMBIN(D1,2))

D1 is my number of decks cell.

For the other ones you need to subtract the flushes.

Quote: miplet

Lets do the easy one first.
Flush

Pick suit combin(4,1)
Pick 5 cards from all of that suit combin(13*d,5)
But the 5 cards can't make a royal or straight flush combin(10,1)*d^5
Or 5 of a kind combin(13,1)*combin(d,5)
Or quads combin(13,1)*combin(d,4)*combin(12,1)*d
Or fullhouse combin(13,1)*combin(d,3)*combin(12,1)*combin(d,2)

Here is the final formula. It works on 5 or more decks. Delete the 5 of a kind portion for less than 5 decks, quads for less than 4 and fullhouse for less than 3.


=COMBIN(4,1)*(COMBIN(13*D1,5)-COMBIN(10,1)*D1^5-COMBIN(13,1)*COMBIN(D1,5)-COMBIN(13,1)*COMBIN(D1,4)*COMBIN(12,1)*D1-COMBIN(13,1)*COMBIN(D1,3)*COMBIN(12,1)*COMBIN(D1,2))

D1 is my number of decks cell.

For the other ones you need to subtract the flushes.

Your flush equations work perfectly.

Many thanks your your help.