Question

Part A Poker Hands: In this activity, we will apply some of the various counting techniques that we have studied including the product and sum rules, the principle of inclusion-exclusion, permutations, and combinations. Our application will be counting the number of ways to be dealt various hands in poker, and analyzing the results.

First, if you are not familiar with poker the following is some basic information.   These are the possible 5-card hands:

Royal Flush (A,K,Q,J,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).

We will 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, 10,…, 2 with four cards removed: 4 of hearts, 8 of spades, 3 of diamonds, and Jack of clubs.   For this game, suppose that you are randomly dealt a 5-card hand from the modified deck.

1.       Count the number of different ways to get each poker hand in your 48 card deck. 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. (Note: if you are unfamiliar with the game you could study the calculations found in many resources on the web before attempting to do these calculations on the modified deck.)

Hand	Counts (48 card deck)
Royal Flush
Straight Flush
Flush
Straight
1 Pair
2 Pair
3-of-a-Kind
4-of-a-Kind
Full House
Nothing

Answer 1