Question

A standard poker deck of 52 cards has four suits, (symbols C, H, S, and D) and thirteen ranks (symbols A, 2, 3, 4, 5, 6, 7, 8, 9, T, J, Q, and K). Every card in the deck has both a value and a suit.1 A poker hand is any set of 5 cards from the standard poker deck. There are some special hands in poker, and these have ranks (i.e. some are better, some are worse). From best to worst, the hands are given names as follows:

(a) A royal flush is a hand T, J, Q, K, A all of the same suit.

(b) A straight flush is a hand of five cards in sequence, all in the same suit.

(c) A four of a kind is a hand with four cards of the same rank.

(d) A full house is a hand with a three cards of one rank and two cards of a different rank.

(e) A flush is a hand with any five cards in the same suit.

(f) A straight is a hand with five cards in sequence.

(g) A three of a kind is a hand with three cards of the same rank.

(h) A two pair is a hand with two different pairs.

(i) A pair is a hand with two cards of the same rank.

(j) A high card hand is a hand which fails all of the above.

1. How many poker hands are there?

2. How many royal flushes are there?

3. How many four of a kinds are there?

4. How many three of a kinds are there which are not a four of a kind and not a full house?

5. How many pairs are there (which are not anything more than a pair)?

6. How many full houses are there?

7. How many two pairs are there which are not full houses?

8. How many straight flushes are there (which are not royal flushes)?

9. How many flushes are there (which are not anything more than a flush)?

10. How many straights are there (which are not anything more than a straight)?

11. How many high card hands are there?

Answer 1

1:

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

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

2:

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.

3:

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 kind is :

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

4:

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