DO NOT ANSWER UNLESS YOU ARE GOING TO ANSWER ALL THREE PARTS OF THE QUESTION Fir

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DO NOT ANSWER UNLESS YOU ARE GOING TO ANSWER ALL THREE PARTS OF THE QUESTION

First, if you are not familiar with poker the following is some basic information. Wewll assume to be using a standard deck containing 52 cards: 4 suits (hearts, clubs, diamonds, and spades) each with the denominations A, K, Q,J,102. For this game, suppose that you are randomly dealt a 5-card hand from the deck. These are the possible 5-card hands: Royal Flus (AKOJ,10 of the same suit); Straight Flush (all 5 cards in order and the same suit) Flush (same suit); Straight (all 5 cards in order); 1 Pair; 2 Pair 3-of-a-Kind; 4-of-a-Kind; Full House (3-of-a-kind and a pair); Nothing (none of the above hands) 1. Count the number of different ways to get each poker hand. The order that you receive the cards does not matter. Show all calculations used to obtain your answers. Do your best to arrive at your answers using counting techniques such as combinations. Do not simply look up the answers on the web. Hand Royal Flush Straight Flush Flush Straight 1 Pair 2 Pair 3-of-a-Kind 4-of-a-Kind Full House Nothing Counts

Explanation / Answer

1. ROyal flush:

no. of suits = 4, only one combination of cards A, K ,Q, J, 10 is possible in each suit.

therefore, no. of ways = 4C1 = 4

2. Straight flush:

One possible sequence = A, 2, 3, 4,5

Starting points possible = A to 9 = 9 ways

no. of suits = 4, hence total no. of ways = 9*4 = 36

3. Flush:

all 5 cards of same suit, and 13 cards in one suit.

no. of ways flush can happen in a suit = 13C5 = 1287 ways

4 suits = 1287*4 = 5148 ways

now, subtracting straight flush and royal flush = 5148 - 36 -4 = 5108 ways

4. Straight:

All cards from the deck can be in a sequence.

So no. of starting points goes from A to 10 = 10 ways

Now since 4 suits each card in the sequence can happen in 4 ways, for example A,2,3,4,5 can happen in 4 ways each card from different suit.

4*4*4*4*4= 1024 ways. 10 starting points = 1024*10 =10240 ways

subtracting royal flush and straught flush = 10240 -36 -4 = 10200 ways

5. 1 pair:

For example: AA234

Ace can happen in 4C2 ways, 2,3,4 can happen in 4C1 ways each. = 4C2 4C14C14C1 = 6*4*4*4 = 384 ways

Now, 2,3,4 can any cards in the except Ace = 12C3= 220 ways

Ace can be any of the 13 cards in deck = 13C1 = 13 ways

Final ans for 1 pair = 384*220*13 = 1098240 ways

6. 2 pair:

example: AA223

Similar to above: 4C24C244C1 = 6*6*44 = 1584 ways.................since last card can be anything of the rest 44 cards

the two cards for pairs can happen in 13C2 ways = 78 ways

total ways for 2 pairs = 1584*78 = 123552 ways

7. 3 of a kind:

example: AAA23

Ace can happen in 4C3 ways and 2,3 in 4C1 ways each = 4*4*4 = 64 ways

Now 2,3 can be anything of the rest 12 cards expect Ace = 12C2 = 66 ways

Ace can be any card of the 13 in a suit = 13C1 =13 ways

Total ways for 3 of a kind = 13*66*64 = 54912 ways

8. 4 of a kind:

example: AAAA2

Ace can happen in only one way, and the last card can be anything from rest 48 cards = 4C448C1 = 48 ways

Ace can be anything of the 13 = 13 ways

Total 4 of a kind = 13*48 = 624 ways

9. full house:

example: AAA22

Ace can happen in 4C3 ways and 2 in 4C2 ways = 4*6 = 24 ways

There are 13(12) permutations of 2 numbers where we get 3 of one and 2 of the other

total full house ways = 24*13*12 = 3744 ways

10. Nothing:

Total no. of possible hands is 52C5 = 2598960 ways

Subtracting all the possible sequences from above = 2598960 - 4 -36 -5108-10200-1098240-123552-54912-624-3744 = 1302540 ways

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