Problem 10 - Algebra 2

Question

Six cards are drawn from a standard deck of cards. How many hands will contain three hearts and three spades?

Step-by-Step Solution

Answer

There are 81,796 different hands with 3 hearts and 3 spades.

Step 1: Understand the problem

We need to determine how many ways we can draw three hearts and three spades from a standard deck of 52 cards. A standard deck has 13 cards of each suit (hearts, spades, diamonds, and clubs).

Step 2: Choose 3 hearts

First, we select 3 cards from the 13 hearts available in the deck. The number of ways to choose 3 hearts from 13 is represented by a combination: \[ \binom{13}{3} \]Calculate this as: \[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]

Step 3: Choose 3 spades

Next, we select 3 cards from the 13 spades available in the deck. Similarly, the number of ways to choose 3 spades from 13 is given by:\[ \binom{13}{3} \]Since we've seen this calculation already, it results in 286 ways: \[ \binom{13}{3} = 286 \]

Step 4: Multiply the combinations

To find the total number of ways to draw 3 hearts and 3 spades, we multiply the number of ways to choose the hearts by the number of ways to choose the spades:\[ 286 \times 286 = 81,796 \]

Step 5: Conclude the solution

Thus, there are 81,796 different hands that consist of 3 hearts and 3 spades.

Key Concepts

Probability

Probability is the branch of mathematics concerned with the study of random events and the likelihood of different outcomes. In the context of drawing cards, probability helps us to understand the chances of getting specific cards from a deck.

• Probability is expressed as a fraction, ratio, or percentage.
• It indicates how likely an event is to occur.

To calculate probability, the number of favorable outcomes is divided by the total number of possible outcomes. For instance, if you want the probability of drawing a heart from a standard deck, you would divide 13 (the number of hearts in a deck) by 52 (the total number of cards). This gives a probability of \(\frac{13}{52} = \frac{1}{4}\), demonstrating that there is a 25% chance of drawing a heart on a single attempt.

Standard Deck of Cards

A standard deck of playing cards consists of 52 cards, which are categorized into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, which are Ace, numbers 2 through 10, and the face cards Jack, Queen, and King.

• The suits are often colored red (hearts and diamonds) and black (clubs and spades).
• The cards are used in a wide variety of games, which focus on strategy, chance, or in some cases, a combination of both.

Understanding the structure of the deck is crucial for solving any problem concerned with drawing cards, as it allows you to estimate probabilities and calculate combinations accurately. In scenarios needing the drawing of a specific number of cards, knowing the card distribution (such as the fact each suit has 13 cards) is essential.

Combinations

In the context of our problem, combinations refer to the number of ways we can select a subset of items from a larger set without considering the order. This is different from permutations, where order does matter.

• The notation for combinations is \(\binom{n}{r}\), which reads as "n choose r".
• This is calculated using the formula: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(!\) indicates factorial, or the product of an integer and all the integers below it.

In the given problem, determining the number of ways to draw 3 hearts from 13 involves calculating \(\binom{13}{3}\). The same applies to drawing 3 spades. Once calculated, these combinations are multiplied to find the total number of successful outcomes, which means picking 3 hearts and 3 spades.

Card Games

Card games are popular as both a social activity and a strategic challenge. They vary greatly in complexity, from simple games like War, which rely entirely on chance, to complex games like Bridge, which require a combination of skill and strategy.

• Card games can have different objectives, like gaining the highest hand value, matching sets, or trick-taking.
• They often involve aspects of probability and combinatorics to determine winning strategies.

In educational contexts, card games can also be an interesting application to understand probability and combinatorics practically. For instance, calculating the number of possible hands or the probability of certain outcomes connects theoretical math with real-world applications, making learning more engaging.