Problem 42 - Algebra 2

Question

Evaluate each expression. \(\frac{5 !}{2 !}\)

Step-by-Step Solution

Answer

The expression evaluates to 60.

Step 1: Understand Factorials

A factorial of a non-negative integer \(n\), represented as \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Compute 5!

Calculate \(5!\) by multiplying all integers from 5 down to 1: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

Step 3: Compute 2!

Calculate \(2!\) by multiplying 2 by 1: \[2! = 2 \times 1 = 2\]

Step 4: Divide Factorials

Substitute the values from Steps 2 and 3 into the original expression and divide: \[\frac{5!}{2!} = \frac{120}{2} = 60\]

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that focuses on using symbols and letters to represent numbers in equations and expressions. Understanding algebra allows students to manipulate equations to solve problems easily. One crucial aspect of algebra is learning to work with mathematical operations like addition, subtraction, multiplication, and division. These operations can involve regular numbers, fractions, and other expressions such as factorials.

Factorials are an essential part of algebraic expressions when dealing with concepts like permutations and combinations. They involve multiplying a series of descending natural numbers. By practicing algebra, students develop problem-solving skills that are applicable in various fields, including physics, engineering, and economics.

• Algebra uses symbols to represent numbers.
• Factorials are frequently used in algebraic expressions.
• Practicing algebra improves overall mathematical skills.

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting, arranging, and combining objects. It plays a significant role in evaluating factorials as seen in the original exercise. Factorials are instrumental in calculations within combinatorics due to their ability to determine the number of ways objects can be arranged or selected.

For instance, the expression \( \frac{5!}{2!} \) often appears in combinatorial problems where one needs to count the number of ways to choose or order objects from a set. This expression can represent permutations or combinations, which are vital concepts in combinatorics.

• Combinatorics focuses on counting and arranging objects.
• Factorials help calculate permutations and combinations.
• Useful in solving problems that involve counting choices or orders.

Mathematics Education

Teaching mathematics effectively is key to helping students build a strong foundation in subjects like algebra and combinatorics. Factorials, a fundamental concept in mathematics, are taught in educational settings to enhance problem-solving abilities. As educators, it is crucial to present mathematical concepts in an engaging and comprehensible manner, starting with basic operations and gradually introducing more complex topics.

Real-life applications can also be integrated into teaching strategies to make concepts like factorials relevant and interesting. By offering a supportive learning environment and using tools such as visual aids and practice problems, students can gain confidence in their mathematical abilities.

• Effective teaching is essential for understanding mathematics.
• Factors like real-life applications make learning engaging.
• Supportive environments foster mathematical confidence.