Problem 53 - Algebra for College Students

Question

How many different seven-card hands can be dealt from a deck of 52 playing cards? \(133,784,560\)

Step-by-Step Solution

Answer

133,784,560 different hands.

Step 1: Understanding the Problem

We need to determine how many different ways we can select 7 cards from a standard deck of 52 playing cards. This is a typical combination problem since the order in which we select the cards doesn't matter.

Step 2: Choosing the Correct Formula

To find how many different groups of 7 cards can be formed from 52 cards, we use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose.

Step 3: Substitute Values into the Formula

In this problem, \(n = 52\) and \(r = 7\). Substituting these values into the combination formula, we get: \[ \binom{52}{7} = \frac{52!}{7!(52-7)!} = \frac{52!}{7!\times45!} \]

Step 4: Computing the Factorial Values

Compute \(52!\), \(7!\), and \(45!\). However, since we're dividing these factorials, we don't actually compute the entire \(52!\). Instead, write out the necessary terms to cancel terms: \[ 52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \] and divide by the factorial of 7: \[ 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 5: Calculate the Result

Perform the necessary multiplications and divisions: \[ \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 133,784,560 \]

Step 6: Conclusion

There are 133,784,560 different seven-card hands possible from a standard deck of 52 cards.

Key Concepts

Combination formula

In combinatorics, when we talk about combinations, we're referring to the selection of items from a larger set where the order does not matter. This is different from permutations where order does matter.
The combination formula is a key formula in combinatorics:

• It is represented as \( \binom{n}{r} \), which is read as "n choose r".
• It calculates the number of ways \( r \) items can be chosen from \( n \) items without regard to order.
• The formula is: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

This formula is very useful when solving problems that involve choosing elements from a set, like determining how many different possible hands can be dealt in a card game.
Using this formula requires knowledge of factorials which allows us to calculate the different possible combinations.

Factorials

Factorials are a mathematical concept represented by an exclamation mark \(!\) and are used frequently in combinations and permutations. A factorial is the product of all positive integers up to a given number. For example, 5 factorial, written as \(5!\), is equal to \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Here are some important points about factorials:

• They provide a way to calculate the total number of ways to arrange a set number of items.
• The factorial of 0, \(0!\), is defined to be 1.
• Factorials grow very quickly with increasing numbers.

In the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), factorials are used to compute the number of combinations possible by cancelling out terms, making it more manageable to calculate large numbers.
Factorials simplify the calculation process significantly when determining the number of potential combinations.

Counting principles

Counting principles refer to the foundational rules and formulas used to determine the number of ways that various events or actions can occur. These principles are often employed in probability and combinatorics.
Some of the critical counting principles include:

• **The Addition Principle:** Used to count the total ways of doing either action by summing individual options.
• **The Multiplication Principle:** Used when you have several events occurring in sequence, and it multiplies the number of choices for each step.

In the case of combinations, the order of the items doesn't matter. This means we use the multiplication principle extensively because multiple actions (like picking a card from a deck) happen in sequence.
To calculate a combination, we multiply the choices for each sequential pick and divide by the arrangement possibilities to ensure the order doesn't matter.
Understanding and applying these counting principles help solve complex problems involving permutations and combinations efficiently.