Problem 39 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(10,4)\)

Step-by-Step Solution

Answer

The value of \( P(10, 4) \) is 5040.

Step 1: Understand the Permutation Formula

To solve a permutation, it's important to know the formula. The permutation formula is given by \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items to arrange.

Step 2: Substitute Values Into the Formula

Given \( P(10, 4) \), substitute \( n = 10 \) and \( r = 4 \) into the permutation formula: \( P(10, 4) = \frac{10!}{(10-4)!} \).

Step 3: Simplify the Factorial Expressions

Calculate \( 10! \) and \( (10-4)! = 6! \).- \( 10! = 10 \times 9 \times 8 \times 7 \times 6! \)- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Cancel and Solve the Equation

The \( 6! \) in the numerator and denominator cancel each other, simplifying the expression to \( 10 \times 9 \times 8 \times 7 \).Calculate the remaining product:\( 10 \times 9 = 90 \)\( 90 \times 8 = 720 \)\( 720 \times 7 = 5040 \)

Step 5: Write the Final Answer

After calculating the above expressions, the final value of \( P(10, 4) = 5040 \).

Key Concepts

Permutation Formula

When you come across the term "permutation" in college algebra, it refers to a specific way of arranging a set of items. Permutations are used when the arrangement matters. There's a handy formula for calculating permutations:

• Formula: \( P(n, r) = \frac{n!}{(n-r)!} \)

Here:

• \( n \) = Total number of items.
• \( r \) = Number of items you want to arrange.

This formula helps determine how many possible ways you can arrange \( r \) items out of a total \( n \). Each position in the sequence makes a difference, which is why it's vital to use the correct order. For example, if you have 10 students and need to arrange 4 as committee leaders, you use the permutation formula \( P(10, 4) \) to find out the number of potential arrangements.

Factorials

Factorials are a fundamental component when working with permutations and combinations in algebra. A factorial, indicated by an exclamation mark (\(!\)), involves multiplying a series of descending natural numbers. For instance:

• \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1 \).

To break it down further, let's look at a specific example using 10 factorial (\( 10! \)). This involves multiplying 10 by each number below it down to 1:

• \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Factorials grow exceptionally fast in size, and they serve a critical role in rearranging and counting combinations in mathematics. When solving permutations like \( P(10, 4) \), you'll calculate factorials to see how many different ways items can be ordered.

Combinatorial Mathematics

Combinatorial mathematics is a branch of mathematics focused on counting, arranging, and analyzing distinct configurations that follow prescribed rules. It's at the heart of combinatorics where both permutations and combinations originate.

• This field allows us to answer questions around how items can be organized, considering different rules and constraints.
• While permutations deal with the order of items, combinations consider how those items can be grouped without regarding the order.

When solving problems such as arranging student leaders, combinatorial mathematics provides the solution through methods like permutations, allowing us to systematically find all possible outcomes. As you dive deeper into this fascinating field, you'll encounter a variety of applications that demonstrate the importance of arrangement and selection in problem-solving.