Evaluate 4! * 3! A Step-by-Step Guide

In this article, we will delve into the evaluation of the mathematical expression 4! * 3!. This expression involves the factorial operation, a fundamental concept in combinatorics and various branches of mathematics. Understanding factorials is crucial for solving problems related to permutations, combinations, and probability. We will break down the expression step by step, explaining the concept of factorials, calculating the individual factorial values, and then multiplying them together to arrive at the final answer. By the end of this exploration, you will have a clear understanding of how to evaluate expressions involving factorials and be equipped to tackle similar problems with confidence.

Understanding Factorials

Before we jump into evaluating the expression, let's first understand what a factorial is. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. In other words, n! = n * (n-1) * (n-2) * ... * 2 * 1. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. The factorial operation is a cornerstone of combinatorics, which deals with counting and arranging objects. It arises naturally in situations where we need to determine the number of ways to arrange a set of items or the number of ways to choose a subset from a larger set. The factorial function grows very rapidly as n increases, making it a powerful tool for counting large numbers of possibilities. Factorials are also used in various other areas of mathematics, such as calculus, where they appear in Taylor series expansions and other formulas. Understanding factorials is essential for anyone working with combinatorial problems or advanced mathematical concepts. In essence, the factorial provides a concise way to represent the product of consecutive integers, making it an indispensable tool in mathematical analysis.

Calculating 4! (4 Factorial)

Now that we have a solid grasp of what factorials are, let's calculate 4!. According to the definition, 4! is the product of all positive integers less than or equal to 4. Therefore, we can write 4! = 4 * 3 * 2 * 1. To evaluate this, we first multiply 4 by 3, which gives us 12. Then, we multiply 12 by 2, resulting in 24. Finally, we multiply 24 by 1, which remains 24. So, 4! = 24. This means that there are 24 different ways to arrange four distinct objects in a sequence. For example, if we have four books, we can arrange them on a shelf in 24 different orders. The calculation of 4! demonstrates the fundamental principle of factorials: multiplying a series of descending integers. This process is straightforward but powerful, allowing us to quickly determine the number of permutations of a small set of objects. The value of 4! is a common one in combinatorial problems, and understanding its calculation is a crucial step in mastering factorial operations. In summary, the calculation 4! = 4 * 3 * 2 * 1 = 24 illustrates the basic mechanics of evaluating a factorial and provides a concrete example of how factorials are used to count arrangements.

Calculating 3! (3 Factorial)

Next, let's calculate 3!. Following the same principle as before, 3! is the product of all positive integers less than or equal to 3. Thus, we have 3! = 3 * 2 * 1. Multiplying 3 by 2 gives us 6, and then multiplying 6 by 1 gives us 6. Therefore, 3! = 6. This result tells us that there are 6 different ways to arrange three distinct objects in a sequence. For instance, if we have three letters, A, B, and C, we can arrange them in 6 different orders: ABC, ACB, BAC, BCA, CAB, and CBA. The calculation of 3! is a simpler example compared to 4!, but it reinforces the concept of factorial calculation. It demonstrates how the factorial function counts the number of permutations of a set of objects, and it highlights the multiplicative nature of the factorial operation. Understanding 3! is essential for grasping more complex factorial calculations and their applications in combinatorics and probability. In essence, 3! = 3 * 2 * 1 = 6 provides a clear illustration of how to calculate factorials for smaller numbers and how these values relate to the number of possible arrangements.

Multiplying 4! and 3!

Now that we have calculated 4! = 24 and 3! = 6, we can proceed to multiply these two values together. The expression we need to evaluate is 4! * 3!, which is equivalent to 24 * 6. Multiplying 24 by 6, we get 144. Therefore, 4! * 3! = 144. This final result represents the product of the number of ways to arrange four objects and the number of ways to arrange three objects. It demonstrates how factorials can be combined through multiplication to solve more complex combinatorial problems. For example, this result could be relevant in a scenario where we need to arrange two separate groups of objects and want to know the total number of ways to arrange both groups. The multiplication of 4! and 3! illustrates the power of factorials in counting and arranging scenarios. In conclusion, 4! * 3! = 24 * 6 = 144 is the final answer, obtained by first calculating the individual factorials and then multiplying them together, showcasing a fundamental operation in combinatorial mathematics.

Final Answer: 144

In summary, we have successfully evaluated the expression 4! * 3!. We began by understanding the definition of a factorial, which is the product of all positive integers less than or equal to a given non-negative integer. We then calculated 4! as 4 * 3 * 2 * 1 = 24 and 3! as 3 * 2 * 1 = 6. Finally, we multiplied these two results together to obtain 24 * 6 = 144. Therefore, the final answer to the expression 4! * 3! is 144. This process demonstrates a fundamental application of factorials in mathematics, particularly in combinatorics and probability. Understanding factorials and their properties is crucial for solving a wide range of problems involving counting, arrangements, and selections. The step-by-step approach we followed—defining factorials, calculating individual factorials, and then combining them—is a standard method for evaluating expressions involving factorials. The result, 144, represents a specific number of arrangements or permutations that can be formed from the combination of two sets of objects. This exercise provides a solid foundation for tackling more complex factorial-based problems in the future, reinforcing the importance of factorials in mathematical reasoning and problem-solving.