Problem 72 - College Algebra with Modeling and Visualization

Question

How many ways are there to draw a 5-card hand from a 52 -card deck?

Step-by-Step Solution

Answer

There are 2,598,960 ways to draw a 5-card hand from a 52-card deck.

Step 1: Understanding the Problem

To find how many ways there are to draw a 5-card hand from a 52-card deck, we need to calculate how many different combinations of 5 cards can be selected from 52 cards. A 5-card hand is a combination since the order of drawing the cards does not matter.

Step 2: Using the Combination Formula

The formula for combinations is given by \( C(n, k) = \frac{n!}{k! (n-k)!} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. In this problem, \( n = 52 \) and \( k = 5 \).

Step 3: Substituting Values into the Formula

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

Step 4: Calculating the Factorials

Calculate the factorials: - \( 52! = 52 \times 51 \times 50 \times 49 \times 48 \times 47! \) - \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) - \( 47! \) cancels out in the denominator and numerator.

Step 5: Simplifying the Expression

After canceling \( 47! \), the expression simplifies to: \[ \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} \] Calculate this expression.

Step 6: Final Calculation

Calculate the simplified expression: - Numerator: \( 52 \times 51 \times 50 \times 49 \times 48 = 311875200 \) - Denominator: \( 120 \) Divide the numerator by the denominator: \[ \frac{311875200}{120} = 2598960 \]

Step 7: Conclusion

Therefore, there are 2,598,960 different ways to draw a 5-card hand from a 52-card deck.

Key Concepts

Combinations

When we talk about combinations, we are referring to a selection of items where the order does not matter. This is different from permutations, where order is important. Combining items without regard to order is called a combination.
The formula for calculating a combination is represented as \( C(n, k) \), which stands for the number of combinations of \( n \) items taken \( k \) at a time. The formula looks like this:

• \[ C(n, k) = \frac{n!}{k!(n-k)!} \]

This formula helps us calculate how many different subsets we can make from a larger set, like picking cards from a deck. In our exercise, we used this principle to find out how many 5-card hands can be drawn from a 52-card deck. Notably, since we do not worry about card order in a hand, combinations perfectly solve this issue.

Factorials

Factorials are a key concept in combinatorics. A factorial of a number \( n \), noted as \( n! \), is the product of all positive integers less than or equal to \( n \).
For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 1! = 1 \)
• By definition, \( 0! = 1 \)

Factorials are vital in the combinations formula because they help in determining the number of possible arrangements of a set. When calculating combinations, factorials help adjust for all the redundant, unnecessary arrangements that arise when the order of selection does not matter. Thus, factorials reduce overcounting of arrangements.

Probability Theory

Probability theory is the study of randomness and uncertainty. It explores how likely events are to happen. When choosing a card, the probability of any outcome can be determined by dividing the number of favorable outcomes by the total number of possible outcomes.
In our problem, we didn't just calculate the probability, but we're stepping into its territory. The exercise used combinations to establish that there are 2,598,960 possible ways to draw 5 cards from a 52-card deck.

• Understanding combinations helps us set up the sample space of card hands.
• Each 5-card hand has an equal likelihood of appearing if you shuffle and draw from the deck randomly.
• Probability would ask 'What's the chance of drawing a particular hand?' and use combinations to solve that.

Thus, probability theory is interwoven with counting techniques like combinations, enabling us to gauge the chance for any specific event within a defined possibility space, such as the deck of cards.