Question

Determine the number of five-card poker hands with the following patterns: (a) Four deuces \(\left(2^{*} s\right)\) and one other card (b) Four of a kind and one other card (c) Two pairs (but not four of a kind) and one card with a different value (d) Three cards of one value and two cards of a second value (this is called a full house) (e) A straight flush (a straight with all the cards from one suit) (f) \(A\) hand with five different card values that is not a straight and is not a flush (g) Three of one kind and two other cards with different values

Short answer

Each pattern has a unique number of poker hands based on combinatorial calculations.

Step-by-step solution

Understanding Four Deuces and One Other Card

For part (a), we focus on getting four deuces. From the four deuces, all of them must be selected, \(\binom{4}{4} = 1\). The one additional card can be any other card except another deuce, giving us 48 remaining cards to choose from \(\binom{48}{1} = 48\) combinations.

Calculating Four of a Kind

In part (b), we're choosing a set of four cards of the same rank. There are 13 possible ranks \(\binom{13}{1} = 13\). Each set can be any card in that rank's four suits. Pick one other card from the 48 remaining cards \(\binom{48}{1} = 48\). Thus, the number of hands is \(13 \times 48\).

Two Pairs and One Other Card

For part (c), select two ranks to form pairs. Choose two pairs from 13 ranks \(\binom{13}{2} = 78\). Choose two suits for each pair: \(\binom{4}{2}^2 = 6 \times 6\). Finally, select a 5th card from the remaining ranks (11 left) \(\binom{11}{1} \binom{4}{1} = 44\). The result is \((78 \times 36 \times 44) \/ 2\).

Calculating a Full House

In part (d), choose a rank for three cards: \(\binom{13}{1} \). Select the suits: \(\binom{4}{3} = 4\). Choose another rank for the pair \(\binom{12}{1}\) and select suits: \(\binom{4}{2} = 6\). Multiply these ( ext{Full House}): \(13 \times 4 \times 12 \times 6\).

Finding a Straight Flush

A straight flush consists of 5 consecutive cards from the same suit. Select a starting rank (1-10), each giving a flush in each of the 4 suits: \(10 \times 4 = 40\).

Non-Straight, Non-Flush Hand

For part (f), select 5 values from 13, \(\binom{13}{5}\). Exclude straights: All cards can't be consecutive. One hand is a flush, so account for straights: 10 hands. Subtract these from total 4 possible suits for each value.

Three of a Kind and Two Singles

In part (g), choose a rank for the three cards: \(\binom{13}{1}\). Pick suits: \(\binom{4}{3} = 4\). Then pick 2 additional cards with different ranks: \(\binom{12}{2}\) and choose suits: \(\binom{4}{1}\times \binom{4}{1} = 16\). Multiply: \(13 \times 4 \times 66 \times 16\).

Key concepts

Card Combinations

Understanding card combinations is a fundamental aspect of combinatorics, especially in games like poker where various hands determine the outcome of the game. In a standard deck of 52 cards, different combinations can be drawn depending on the specific rules being followed. When determining how many ways we can select a certain sequence of cards, it's crucial to think about:

• The number of cards in the combination
• The restrictions placed (e.g., same suit, same rank, etc.)
• The number of available cards in each rank or suit

For instance, drawing four deuces means once those cards are fixed, the fifth card can be any from the remaining 48 cards in the deck, calculated as \( \binom{48}{1} = 48 \). Understanding these specific combinations helps players strategize, potentially predicting opponents' hands based on the visible cards.

Poker Hand Patterns

Poker hand patterns provide a structured way to evaluate a player's hand. They are ranked based on the rarity and difficulty of being drawn. Common patterns include:

• Four of a Kind: Four cards of one rank, minus almost any additional card in the deck, result in a strong poker hand.
• Full House: Consisting of three cards of one rank and two cards of another, the chance of crafting such a hand requires a unique combination of ranks and suits.
• Straight Flush: Five consecutive cards all from the same suit. It's among the rarest hands due to its dual requirements of both sequential rank and uniform suit.

Recognizing these patterns necessitates awareness of the possible card configurations, and understanding these can enhance your ability to calculate potential hands and strategize accordingly.

Binomial Coefficient

The binomial coefficient, symbolized as \( \binom{n}{k} \), plays a significant role in combinatorics, particularly when calculating card combinations. It is used to determine the number of ways to select \( k \) items from a total of \( n \) without regard to order. This formula is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]In the context of poker, the binomial coefficient helps calculate various hands. For instance, to select 2 out of 4 suits for a pair, we use \( \binom{4}{2} \). Similarly, to choose ranks and suits in crafting a specific poker hand (e.g., full house or two pairs), understanding and applying this formula is crucial.
Breaking down complex card combinations into manageable calculations through the binomial coefficient makes solving card game-related problems a lot easier, helping one to gauge probabilities accurately.