Problem 28 - A Graphical Approach to College Algebra

Question

Use a calculator to evaluate each expression. $$32 P_{4}$$

Step-by-Step Solution

Answer

The value of \( 32P_4 \) is 863040.

Step 1: Understand the Notation

The notation \( nP_r \) refers to a permutation, representing the number of ways to arrange \( r \) items from a set of \( n \) items. For example, \( 32P_4 \) signifies the number of ways to arrange 4 items from a total of 32 items.

Step 2: Use the Permutation Formula

The formula for permutations is given by: \[ nP_r = \frac{n!}{(n-r)!} \]In our case, \( n = 32 \) and \( r = 4 \). Substitute these values into the formula.

Step 3: Calculate Factorials

First, calculate the factorial of 32, denoted as \( 32! \), which is the product of all positive integers up to 32. However, the full calculation can be simplified by recognizing that the permutation formula allows cancellation:\( \frac{32!}{(32-4)!} = \frac{32!}{28!} \) which simplifies to the product of reducing factors from 32 down to 29: \( 32 \times 31 \times 30 \times 29 \).

Step 4: Evaluate the Expression

Using a calculator, multiply the products obtained in the previous step:\( 32 \times 31 = 992 \)\( 992 \times 30 = 29760 \)\( 29760 \times 29 = 863040 \)Therefore, \( 32P_4 = 863040 \).

Key Concepts

Permutation Formula

Permutations are all about arranging items in specific sequences, which is a common requirement in algebra and real-life scenarios. The notation \( nP_r \) expresses this, where \( n \) stands for the total number of items, and \( r \) is the number of items to arrange.
The permutation formula helps us calculate these arrangements and is given by:

• \( nP_r = \frac{n!}{(n-r)!} \)

This formula considers both selection and order, which differentiates it from combinations, where order doesn't matter. In the example \( 32P_4 \), you are asked to determine how many ways you can arrange 4 items from 32. Understanding this concept is crucial as it not only assists in solving algebraic problems involving permutations but also provides a foundation for more complex probability and statistics problems. By substituting the values, you can see how the formula simplifies large factorials, making complex calculations manageable.

Factorials

Factorials are a fundamental concept in mathematics, especially in permutations and combinations. Represented by an exclamation mark (!), it indicates the product of all positive integers up to a specified number.
For instance, \( n! \) is calculated by multiplying all whole numbers from 1 to \( n \). In our permutation exercise, finding \( 32! \) directly is impractical due to its large size.

• Instead, you calculate the factorial of \( n-r \), where \( r \) is the number of items to arrange.
• This involves simplifying using \( \frac{n!}{(n-r)!} \), which reduces the calculation as larger factorials cancel out.

For \( 32P_4 \), this becomes \( \frac{32!}{28!} \), simplifying to \( 32 \times 31 \times 30 \times 29 \). Factorials are essential for simplifying these large calculations and understanding various algebraic expressions related to arrangement problems.

Calculator Use in Algebra

Using a calculator efficiently in algebra can significantly simplify solving complex permutation problems. With large numbers like those in factorials, calculators help prevent computational errors and save time.
For evaluating expressions such as \( 32P_4 \), the process involves breaking down the problem:

• First, simplify the expression with the permutation formula, reducing large factorials.
• Next, multiply the relevant numbers sequentially, using multiplication functions on calculators.

For the problem at hand:

• Multiply the sequence: \( 32 \times 31 \), then multiply the result by 30, and finally by 29.
• These successive calculations can be quickly performed using the calculator’s memory functions, minimizing mistakes.

By effectively using a calculator, you bridge complicated algebraic expressions with straightforward arithmetic, enhancing your problem-solving skills and efficiency.