Question

Explain where all the numbers in the Computations of the Poker Hands come from.

Answer 1

The royal flush means 5 cards ten , jack, queen, king, and ace of one suit. There are only 4 ways to select royal flush. So number of possible royal flush is 4.

-----------

A straight flush means all 5 cards are from same suit and they form a straight. Since number of possible straight from each suit is 10 and number of ways of selecting one suit out of 4 is C(4,1) =4 so possible number of straight flush is 10*4 = 40.

-------------------

Flush:

There are total 4 suits and each suit has 13 cards. So number of ways of selecting 1 suit and then 5 cards out of 13 is

$\binom{4}{1}\binom{13}{5}=5148$

-----------------------

Straight can be start from 10 cards start from ace, 1, 2,3, 4...10. Each denomination can be choosed in C(4,1)= 4 ways. So number of possible straights is

$10\cdot C(4,1)\cdot C(4,1)\cdot C(4,1)\cdot C(4,1)\cdot C(4,1)=10240$

From the above, we need to remove flush. A straight flush means all 5 cards are from same suit and they form a straight. Since number of possible straight from each suit is 10 and number of ways of selecting one suit out of 4 is C(4,1) =4 so possible number of straight flush is 10*4 = 40.

Therefore number of straight without being a flush is 10240 - 40 = 10200

---------------------

1 Pair:

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 1 denomination and then 2 cards out of 4 is

$\binom{13}{1}\binom{4}{2}$

And since we need exactly 1 pair so remaining 3 cards must come from different denominations so number of ways of selecting 3 denominations out of remaining 12 denominations and then 1 card from each selected denominations is

$\binom{12}{3}\binom{4}{1}\binom{4}{1}\binom{4}{1}$

So number of ways of selecting 1 pair is :

$\binom{13}{1}\binom{4}{2}\binom{12}{3}\binom{4}{1}\binom{4}{1}\binom{4}{1}=1098240$

---------------------

2 pair:

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 2 denominations and then 2 cards out of 4 is

$\binom{13}{2}\binom{4}{2}\binom{4}{2}$
And since we need exactly 2 pairs so remaining 1 card must come from different denomination so number of ways of selecting 1 denominations out of remaining 11 denominations and then 1 card from selected denomination is

$\binom{11}{1}\binom{4}{1}$
So number of ways of selecting 2 pairs is :

$\binom{13}{2}\binom{4}{2}\binom{4}{2}\binom{11}{1}\binom{4}{1}=123552$

--------------------

Four of a kind:

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 1 denominations and then 4 cards out of 4 is

$\binom{13}{1}\binom{4}{4}=13$
And since we need 4 of same kind so remaining 1 card must come from different denomination so number of ways of selecting 1 denominations out of remaining 12 denominations and then 1 card from selected denomination is

$\binom{12}{1}\binom{4}{1}=48$
So number of ways of selecting four of a kindis :

$\binom{13}{1}\binom{4}{4}\binom{12}{1}\binom{4}{1}=13\cdot 48=624$

-------------------

3 of a kind:

Number of ways of selecting 1 denominations out of 13 is C(13,1). Number of ways of selecting 3 cards out of 4 cards of selected denomination is C(4,3). And then select two denominations out of remaining 12 denominations is C(12,2) and then 1 card from each selected denominations is C(4,1)C(4,1). So number of ways are there to draw a 5 card poker hand that contains 3 a kind is

C(13,1)C(4,3)C(12,2)C(4,1)C(4,1) = 54912 ways

--------------------

Full house:

Number of ways of selecting 1 denominations out of 13 is C(13,1). Number of ways of selecting 3 cards out of 4 cards of selected denomination is C(4,3). And then select one denomination out of remaining 12 denominations is C(12,1) and then 2 cards from each selected denominations is C(4,2). So number of ways are there to draw a 5 card poker hand that contains 3 a kind and 2 a kind is

C(13,1)C(4,3)C(12,1)C(4,2) = 3744 ways

----------------------

Sum of all above hands: 1296462

Number of ways of selecting 5 cards out of 52 cards is

$\binom{52}{5}=2598960$

Number of ways of getting no poker hand

Nothing: 2598960 - 1296462 = 1302498