Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get a full house? (A full house consists of three of one kind and two of another. For example, \(8-8-8-5-5\) and \(\mathrm{K}-\mathrm{K}-\mathrm{K}-10-10\) are full houses.)

Step-by-Step Solution

Answer

There are 1872 ways to get a full house from a deck of 52 cards.

Step 1: Determine Ways to Select 2 Types of Cards

The first step is to figure out in how many ways we can select two different types of cards (for instance, eights and fives). As there are 13 types of cards in a deck (Ace through King), this can be achieved in \( \binom{13}{2} \) ways, where \( \binom{n}{r} \) is a binomial coefficient that provides the number of possible combinations of n items taken r at a time.

Step 2: Determine Ways to Select 3 Cards of 1 Type and 2 Cards of the Other Type

Next, we need to calculate how many ways we can select 3 cards of one kind and 2 cards of another kind. Since each type of card is represented in 4 suits, this is done in \( \binom{4}{3} \times \binom{4}{2} \) ways.

Step 3: Apply the Multiplication Principle

Now, we use the multiplication principle. This states that if event A can happen in m ways and event B can happen in n ways, then event A happening AND event B happening can happen in \( m \times n \) ways. Thus, there are \( \binom{13}{2} \times \binom{4}{3} \times \binom{4}{2} \) ways to get a full house from a deck of 52 cards.

Step 4: Calculate the Results

We calculate these values to get the result. \( \binom{13}{2} = 78 \), \( \binom{4}{3} = 4 \), and \( \binom{4}{2} = 6 \). So, the total number of ways to get a full house is \( 78 \times 4 \times 6 = 1872 \).

Key Concepts

Binomial Coefficient

When we talk about the binomial coefficient in combinatorics, we are referring to a way to determine how many ways we can choose a certain number of items from a larger set.
The mathematical notation for this is \( \binom{n}{r} \), pronounced "n choose r." Here, \( n \) represents the total number of items to choose from, and \( r \) represents the number of items we want to choose.
The formula for calculating the binomial coefficient is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Let's break this down further:- \(!\) is the factorial symbol, meaning you multiply the number by all the integers less than it down to one.
For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.- In our original exercise, we used \( \binom{13}{2} \). This tells us how many ways we can choose 2 card types from 13 available types (Ace through King) in a standard deck.
Using the formula, \( \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 \).
This calculation tells us there are 78 ways to select two different ranks of cards.

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics that helps us determine the number of different combinations of events. It states that if one event can occur in \( m \) ways and a subsequent event in \( n \) ways, then both events together can occur in \( m \times n \) ways.

In the context of our exercise, once we select the two ranks of cards (say, sevens and jacks), we need to determine how we can arrange 3 cards of one type and 2 cards of another from the respective selected ranks.

• There are \( \binom{4}{3} \) ways to choose 3 cards from 4 available (like 4 sevens in different suits).
• Similarly, there are \( \binom{4}{2} \) ways to choose 2 out of 4 cards from the other rank.

Using the multiplication principle, we combine all these steps:
- First select 2 ranks of cards from 13: \( 78 \) ways.- Choose 3 of one type: \( 4 \) ways.- Choose 2 of the other type: \( 6 \) ways.- Therefore, the total is \( 78 \times 4 \times 6 = 1872 \) ways to form a full house.

Probability

Probability involves determining the likelihood of a particular outcome when there are multiple possible outcomes. In card games, calculating probability can be quite interesting due to the vast number of possible card combinations.
For any event, the probability is calculated as:

• \( P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

In our scenario with the full house, to determine the probability of drawing a full house from a standard deck:- We know there are \( 1872 \) favorable ways to get a full house.- The total number of 5-card hands is \( \binom{52}{5} = \frac{52!}{5! (47!)} = 2,598,960 \).Using the probability formula, the chance of getting a full house in a single 5-card draw is:\[P = \frac{1872}{2,598,960} \approx 0.00072\]This is roughly a 0.072% chance, highlighting the rarity of being dealt such a hand in a game!