Problem 45 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Draw Poker Hands How many different five-card hands can be dealt from a deck of 52 cards?

Step-by-Step Solution

Answer

There are 2,598,960 different five-card poker hands possible from a 52-card deck.

Step 1: Understand the Problem

The problem asks us to calculate the number of different combinations of five cards that can be drawn from a standard deck of 52 cards. This is a typical combinations problem where the order of the cards does not matter.

Step 2: Identify the Formula for Combinations

The number of combinations of selecting \( r \) objects from a set of \( n \) objects is given by the formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] In this problem, \( n = 52 \) and \( r = 5 \) since there are 52 cards in the deck and we are choosing 5 cards.

Step 3: Plug Values into the Combinations Formula

Substitute \( n = 52 \) and \( r = 5 \) into the combinations formula: \[ C(52, 5) = \frac{52!}{5!(52-5)!} = \frac{52!}{5! imes 47!} \]

Step 4: Simplify the Factorial Expression

Using factorial simplification, only expand the necessary part of the factorial to reduce the expression: \[ C(52, 5) = \frac{52 imes 51 imes 50 imes 49 imes 48}{5 imes 4 imes 3 imes 2 imes 1} \]

Step 5: Calculate Numerator and Denominator

Calculate the numerator: \[ 52 imes 51 imes 50 imes 49 imes 48 = 311,875,200 \] Calculate the denominator: \[ 5 imes 4 imes 3 imes 2 imes 1 = 120 \]

Step 6: Divide to Find the Number of Combinations

Now, divide the numerator by the denominator to find the number of combinations: \[ \frac{311,875,200}{120} = 2,598,960 \] This is the number of different five-card poker hands possible from a 52-card deck.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics, dealing with the selection of items from a larger set where the order does not matter. Imagine you have a basket of fruit, and you need to pick a certain number without caring about the sequence they are picked in. That's where combinations come into play. The mathematical formula for combinations is:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here, \( n \) is the total number of items to choose from, \( r \) is the number of items you want to choose, and \(!\) represents the factorial function.
This formula helps us calculate the number of ways we can choose \( r \) items from \( n \) total items. By applying this to our problem, we can determine how many distinct five-card hands are possible from a deck of 52 cards.

Factorial

The factorial function, denoted by \(!\), is an essential operation in combinatorics. It refers to the product of all positive integers up to a given number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
This concept is critical in the combinations formula since both the numerator and the denominator include factorial expressions. Factorial allows us to calculate arrangements and selections efficiently, simplifying complex counting problems. In our exercise, it helps divide the large number of total permutations by the permutations that do not need to be counted, giving us the number of combinations.

• For combination issues: \( n! \) represents all possible arrangements of \( n \) elements.
• \( r! \) adjusts for the subset we're choosing, and \( (n-r)! \) for all remaining elements not chosen.

With factorial, large number calculations become manageable in counting methods.

Poker Hands

Poker hands consist of five cards dealt from a full deck. In games like poker, the hand you hold determines your chances of winning based on predefined rules of card rankings. As there are various possible combinations in a poker hand, understanding how to calculate these combinations ensures you comprehend the probabilities in games of chance.
In the original exercise, the question is focused on how many different five-card hands we can have. Each combination represents a unique poker hand with different possibilities of ranks and suits. Calculating these possibilities ensures that players have an understanding of the likelihood and strategy involved, whether they are playing for fun or strategizing for a tournament.
This knowledge is specifically useful when evaluating the strength of your hand or predicting possible outcomes during a round of poker.

Deck of Cards

A standard deck of cards contains 52 cards, divided into four suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards, ranging from ace to king.
Understanding the composition of a deck is crucial in solving problems involving cards. With knowledge of how many total cards exist and the distribution across ranks and suits, you can accurately compute combinations and probabilities for card games like poker.
When calculating poker hands, knowing there are 52 unique cards means any mix of five cards can form a potential hand. This context provides foundational insight into counting and probability used in card games and other probability-based queries.
Grasping these basics fosters better comprehension of not only card games but also related probability problems that revolve around secure understanding of a standard deck.