Solving Factorial Expressions Finding The Value Of 10!/(10-3)!

#Find the value of $rac{10!}{(10-3)!}$ can seem daunting at first, but with a clear understanding of factorials and a systematic approach, it becomes a manageable and even enjoyable mathematical exercise. In this comprehensive guide, we will embark on a journey to demystify factorial expressions, unravel the intricacies of the given problem, and arrive at the correct solution with confidence. We will delve into the fundamental concepts of factorials, explore their properties, and apply them to the specific expression at hand. By the end of this exploration, you will not only be able to solve this particular problem but also gain a deeper appreciation for the power and elegance of factorial notation in mathematics.

Factorials, at their core, represent the product of all positive integers less than or equal to a given non-negative integer. This seemingly simple concept forms the bedrock of various mathematical disciplines, including combinatorics, probability, and calculus. Understanding factorials is crucial for counting permutations, combinations, and probabilities, as well as for working with series and functions in calculus. The notation "n!" represents the factorial of a non-negative integer "n," and it is defined as the product of all positive integers from 1 up to n. For example, 5! (read as "5 factorial") is equal to 5 * 4 * 3 * 2 * 1, which equals 120. The factorial of 0 is defined as 1, which might seem counterintuitive at first, but it is a necessary convention for consistency in various mathematical formulas and theorems. Mastering the concept of factorials opens doors to a wide range of mathematical applications and problem-solving techniques.

To effectively solve factorial expressions, it is essential to grasp their key properties. One of the most important properties is the recursive nature of factorials, which means that the factorial of a number can be expressed in terms of the factorial of a smaller number. Specifically, n! = n * (n-1)!. This recursive relationship allows us to simplify factorial expressions by breaking them down into smaller, more manageable parts. For example, 10! can be written as 10 * 9!, which can further be broken down as 10 * 9 * 8!, and so on. This property is particularly useful when dealing with fractions involving factorials, as it often allows for cancellations and simplifications. Another important property is that factorials grow very rapidly as n increases. This rapid growth makes factorials essential in counting problems where the number of possibilities explodes exponentially. Understanding these properties empowers us to manipulate factorial expressions with greater ease and efficiency, paving the way for solving complex mathematical problems.

Decoding the Expression: 10!/(10-3)!

Now, let's turn our attention to the specific problem at hand: finding the value of $rac{10!}{(10-3)!}$. This expression involves the ratio of two factorials, which is a common scenario in various mathematical contexts. To solve this, we need to carefully expand the factorials and identify opportunities for simplification. The numerator, 10!, represents the factorial of 10, which is the product of all positive integers from 1 to 10. The denominator, (10-3)!, represents the factorial of 7, which is the product of all positive integers from 1 to 7. To evaluate this fraction, we can expand both factorials and look for common factors that can be cancelled out. This process of expansion and cancellation is a fundamental technique in simplifying factorial expressions and arriving at the final solution. By systematically applying this technique, we can transform the seemingly complex expression into a more manageable form, ultimately leading us to the correct answer.

To begin the simplification process, let's expand the factorials in the expression $rac{10!}{(10-3)!}$. We know that 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 and (10-3)! = 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1. Now, we can rewrite the expression as $rac{10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1}{7 * 6 * 5 * 4 * 3 * 2 * 1}$. Notice that the denominator, 7!, is a part of the numerator, 10!. This observation is crucial because it allows us to cancel out the common factors in the numerator and denominator, significantly simplifying the expression. By cancelling out the common factors, we eliminate the need to compute the entire factorial of 10, which would be a much larger number. This cancellation technique is a powerful tool in simplifying factorial expressions and making them easier to evaluate.

The Art of Cancellation: Simplifying Factorial Fractions

After expanding the factorials, we can observe a remarkable opportunity for simplification. The expression $rac{10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1}{7 * 6 * 5 * 4 * 3 * 2 * 1}$ clearly shows that the terms from 7 down to 1 appear in both the numerator and the denominator. This means we can cancel out these common factors, leaving us with a much simpler expression. Cancelling out the 7 * 6 * 5 * 4 * 3 * 2 * 1 terms from both the numerator and the denominator, we are left with $rac{10 * 9 * 8}{1}$. This simplification significantly reduces the complexity of the calculation, making it easier to arrive at the final answer. The art of cancellation is a fundamental technique in simplifying factorial expressions, and it highlights the elegance and efficiency of mathematical notation. By recognizing and exploiting opportunities for cancellation, we can transform complex expressions into more manageable forms, ultimately making problem-solving a more streamlined and enjoyable process.

Now that we have cancelled out the common factors, we are left with the simplified expression 10 * 9 * 8. This expression represents the product of three consecutive integers, which is a common pattern in factorial problems. To find the value of this expression, we simply multiply the numbers together. First, we can multiply 10 and 9, which gives us 90. Then, we multiply 90 by 8, which gives us 720. Therefore, the value of the simplified expression 10 * 9 * 8 is 720. This calculation demonstrates how simplifying factorial expressions through cancellation can transform a seemingly complex problem into a straightforward multiplication. The ability to simplify and manipulate mathematical expressions is a crucial skill in mathematics, and it allows us to tackle a wide range of problems with confidence and efficiency.

The Grand Finale: Arriving at the Solution

Having simplified the expression $rac{10!}{(10-3)!}$ to 10 * 9 * 8, we have arrived at the final step: calculating the product. As we determined in the previous section, 10 * 9 * 8 equals 720. Therefore, the value of the original expression is 720. This result corresponds to option C in the given choices. The journey from the initial factorial expression to the final numerical answer has been a testament to the power of simplification and systematic problem-solving. By understanding the properties of factorials, expanding the expressions, cancelling out common factors, and performing the final multiplication, we have successfully navigated the problem and arrived at the correct solution. This process not only provides the answer to this specific problem but also equips us with valuable skills and techniques for tackling other mathematical challenges involving factorials and similar concepts.

In conclusion, the value of $rac{10!}{(10-3)!}$ is 720, which corresponds to option C. This problem serves as a valuable illustration of the importance of understanding factorials, their properties, and the techniques for simplifying factorial expressions. By mastering these concepts, we can confidently tackle a wide range of mathematical problems involving permutations, combinations, and other related topics. The ability to manipulate and simplify mathematical expressions is a crucial skill in various fields, from science and engineering to finance and computer science. As we continue our mathematical journey, the principles and techniques learned in solving this problem will undoubtedly prove invaluable in our future endeavors. The world of mathematics is full of fascinating challenges and rewarding discoveries, and the journey of exploration and learning is an adventure in itself.

Therefore, the correct answer is C. 720