Evaluating Factorial Expressions Simplifying 5!2! / 6!4!
In the realm of mathematics, factorial expressions often appear in various contexts, such as combinatorics and probability. These expressions involve the product of consecutive integers down to 1, and they play a crucial role in counting and arrangement problems. In this article, we delve into the evaluation of a specific factorial expression, aiming to simplify it as much as possible. Our focus will be on the expression 5!2! / 6!4!, where the exclamation mark denotes the factorial operation. We will meticulously break down the factorials, identify common factors, and perform cancellations to arrive at the simplest form of the expression. This process not only enhances our understanding of factorial manipulation but also provides valuable insights into the techniques used to simplify complex mathematical expressions. The ability to efficiently simplify factorial expressions is a fundamental skill in mathematics, enabling us to tackle more intricate problems with confidence and precision. As we embark on this journey of simplification, we will uncover the elegance and power of mathematical notation, demonstrating how it allows us to express complex ideas in a concise and understandable manner. Our goal is to not only find the numerical answer but also to illustrate the underlying principles that govern factorial operations, ensuring a comprehensive and enriching learning experience. By the end of this article, you will be equipped with the knowledge and skills necessary to approach similar factorial expressions with ease and accuracy.
Understanding Factorials
Before diving into the simplification, it's essential to have a solid grasp of what factorials represent. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Mathematically, this is expressed as n! = n × (n-1) × (n-2) × ... × 2 × 1. For instance, 5! (read as "5 factorial") is 5 × 4 × 3 × 2 × 1 = 120. Factorials are central to combinatorics, where they count the number of ways to arrange a set of distinct objects. For example, 5! represents the number of ways to arrange 5 distinct items in a sequence. The concept of factorials extends beyond simple counting problems, appearing in calculus, statistics, and other branches of mathematics. Understanding factorials is crucial for solving permutation and combination problems, where the order of selection and arrangement matters. In the context of our problem, we have multiple factorials involved, each representing a different product of integers. To simplify the expression 5!2! / 6!4!, we need to expand these factorials and identify common factors that can be canceled out. This process requires a careful examination of the numbers involved and a systematic approach to simplification. By breaking down the factorials into their constituent products, we can reveal the underlying structure of the expression and make it easier to manipulate. The efficient calculation of factorials and their simplification are key skills in mathematics, allowing us to solve complex problems with relative ease. As we proceed with the simplification, we will see how a thorough understanding of factorials can lead to elegant and concise solutions.
Expanding the Factorials
To begin the simplification process, we need to expand each factorial in the expression 5!2! / 6!4!. Let's start by writing out the expansions:
- 5! = 5 × 4 × 3 × 2 × 1
- 2! = 2 × 1
- 6! = 6 × 5 × 4 × 3 × 2 × 1
- 4! = 4 × 3 × 2 × 1
Now, we can substitute these expansions back into the original expression:
5!2! / 6!4! = (5 × 4 × 3 × 2 × 1) × (2 × 1) / (6 × 5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1)
This expansion reveals the detailed structure of the expression, allowing us to identify common factors in the numerator and the denominator. The presence of common factors is the key to simplifying the expression, as they can be canceled out, reducing the overall complexity. Expanding the factorials is a crucial step in this process, as it transforms the expression from a symbolic representation into a concrete product of integers. This transformation makes it easier to see the relationships between the numbers and to apply simplification techniques. The careful expansion of factorials is essential for avoiding errors and ensuring an accurate simplification. By writing out each factorial in its expanded form, we gain a clearer understanding of the expression and pave the way for the next step in the simplification process. This methodical approach is a hallmark of mathematical problem-solving, where breaking down complex problems into smaller, manageable steps is key to success. As we move forward, we will see how the expanded form of the expression allows us to identify and cancel out common factors, leading us closer to the final simplified answer.
Identifying and Canceling Common Factors
With the factorials expanded, we can now focus on identifying and canceling common factors in the numerator and the denominator. Looking at the expanded expression:
(5 × 4 × 3 × 2 × 1) × (2 × 1) / (6 × 5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1)
We can observe several common factors. The term (5 × 4 × 3 × 2 × 1) appears in both the numerator and the denominator, allowing us to cancel it out entirely. Additionally, we have a (2 × 1) term in the numerator. In the denominator, we have (4 × 3 × 2 × 1), which is 4!. We can rewrite the expression to highlight these cancellations:
(5!) × (2!) / (6!) × (4!) = (5!) / (6!) × (2!) / (4!)
Now, we can express 6! as 6 × 5! and 4! as 4 × 3 × 2 × 1. This allows us to further simplify the expression:
(5!) × (2 × 1) / (6 × 5!) × (4 × 3 × 2 × 1)
Canceling out the common factor of 5! gives us:
(2 × 1) / (6) × (4 × 3 × 2 × 1) = 2 / (6 × 24)
Further simplification involves canceling common factors between 2 and 6, resulting in:
1 / (3 × 24)
This leaves us with the simplified expression:
1 / 72
The process of canceling common factors is a fundamental technique in simplifying fractions and expressions. It relies on the principle that dividing both the numerator and the denominator by the same non-zero number does not change the value of the fraction. In this case, we systematically identified and canceled common factors, reducing the complexity of the expression step by step. This approach not only leads to the correct answer but also demonstrates the power of mathematical simplification techniques. By carefully examining the expanded factorials, we were able to reveal the underlying structure of the expression and apply the necessary cancellations. This skill is essential for tackling more complex mathematical problems and is a testament to the elegance and efficiency of mathematical notation.
Final Simplification and Result
After canceling the common factors, we arrived at the expression 1 / (3 × 24). To complete the simplification, we need to perform the multiplication in the denominator:
3 × 24 = 72
Therefore, the final simplified form of the expression is:
1 / 72
This result represents the simplest form of the original expression 5!2! / 6!4!. The process of simplification involved expanding the factorials, identifying common factors, and canceling them out systematically. By following these steps, we were able to reduce the expression to its most basic form, demonstrating the power of mathematical simplification techniques. The final answer, 1 / 72, is a fraction that cannot be further simplified, as the numerator and denominator have no common factors other than 1. This result is a testament to the elegance and precision of mathematics, where complex expressions can often be reduced to simple and understandable forms. The skill of simplifying expressions is a valuable asset in mathematics, allowing us to solve problems more efficiently and accurately. By mastering these techniques, we can approach mathematical challenges with confidence and clarity. The journey from the original expression to the final simplified answer has highlighted the importance of understanding factorials, identifying common factors, and applying simplification rules. This comprehensive approach is crucial for success in mathematics and beyond.
In conclusion, the evaluation and simplification of the expression 5!2! / 6!4! led us to the final answer of 1 / 72. This process involved several key steps, including expanding the factorials, identifying common factors in the numerator and denominator, and canceling these factors to arrive at the simplest form. The ability to simplify factorial expressions is a fundamental skill in mathematics, with applications in various fields such as combinatorics, probability, and calculus. Understanding factorials and how to manipulate them is crucial for solving a wide range of mathematical problems. The techniques used in this simplification, such as identifying and canceling common factors, are applicable to many other mathematical contexts. By mastering these skills, we can approach complex problems with confidence and precision. The journey from the initial expression to the final simplified answer has demonstrated the power of mathematical notation and the elegance of simplification techniques. The result, 1 / 72, is a concise and understandable representation of the original expression, highlighting the beauty and efficiency of mathematics. This exercise serves as a valuable example of how mathematical problems can be approached systematically, leading to clear and accurate solutions. The principles and techniques discussed in this article can be applied to a variety of mathematical challenges, making them an essential part of any mathematical toolkit. As we continue our exploration of mathematics, the skills gained from this exercise will serve as a solid foundation for tackling more advanced topics and problems.
