Problem 14 - Algebra 2

Question

Evaluate each expression. \(P(12,6)\)

Step-by-Step Solution

Answer

The value of \(P(12,6)\) is 924.

Step 1: Understand the Formula

To evaluate permutations, we use the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]where \(n\) is the total number of items, and \(r\) is the number of items to arrange.

Step 2: Substitute Values

For this problem, we have \(n = 12\) and \(r = 6\). Substituting these into the formula gives:\[ P(12, 6) = \frac{12!}{(12-6)!} \]

Step 3: Simplify the Expression

Calculate the factorial expressions:- Calculate \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)- Calculate \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\) Now the formula becomes:\[P(12, 6) = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}\]

Step 4: Calculate the Numerator and Denominator

Now compute both the numerator and denominator:- The numerator: \(12 \times 11 \times 10 \times 9 \times 8 \times 7 = 665280\)- The denominator: \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

Step 5: Divide to Find the Permutation

Divide the numerator by the denominator to get the result:\[P(12, 6) = \frac{665280}{720} = 924\]

Key Concepts

Factorial

The factorial of a number is a fundamental operation in mathematics, especially in permutations and combinations. It is denoted by an exclamation mark, for example, the factorial of 12 is written as 12!. But what does that mean exactly?

Essentially, the factorial of a number \(n\) is the product of all positive integers up to that number. So, 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1. That is the core concept of factorial: a long multiplication chain for whole numbers.

Keep in mind:

• The factorial of 0 is defined as 1: \(0! = 1\)
• Factorials are only defined for non-negative integers.
• The values grow extremely fast with increasing numbers.

Understanding the factorial function is key for grasping more advanced concepts like permutations.

Combinatorics

Combinatorics is an area of mathematics primarily about counting, arranging, and finding patterns. A significant part of combinatorics is the study of permutations. Permutations, such as those denoted by \(P(n, r)\), refer to the different ways of arranging \(r\) items from a total of \(n\) items.

Here's a breakdown of some key ideas:

• **Permutations** consider the order, which means rearranging elements gives different permutations.
• The permutation formula \(P(n, r) = \frac{n!}{(n-r)!}\) calculates the number of such arrangements.
• Factorials are used in this formula to simplify the calculation of large product terms.

Combinatorics is crucial for fields like computer science, cryptography, and game theory, as it provides the tools for solving problems related to arrangements and probabilities.

Algebra 2

Algebra 2 is often where students first encounter advanced topics like permutations. Algebra 2 builds on Algebra 1 by introducing new concepts and refining students' problem-solving skills. Permutations often come up in this course as real-life applications of algebraic thinking. Let's look at how Algebra 2 connects to permutations:

• **Problem Solving**: Permutations are used to solve problems that involve arranging objects.
• **Use of Equations**: Algebra 2 teaches how to manipulate equations, like those used for permutations \(P(n, r)\).
• **Exploration of Functions**: Factoring and expanding expressions, which are techniques students practice, align with utilizing factorials for permutations.

By understanding permutations, students can apply algebraic rules to solve complex problems, enhancing their logical and analytical skills.