How To Calculate 10! Divided By (10-3)! A Step-by-Step Solution

#Problem

We are asked to find the value of the expression $rac{10!}{(10-3)!}$. This problem involves the concept of factorials, which is a fundamental topic in combinatorics and mathematics. To solve this, we need to understand what a factorial means and how to simplify expressions involving factorials.

Understanding Factorials

A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Mathematically, it is defined as:

n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1

For example:

• 5*! = 5 × 4 × 3 × 2 × 1 = 120
• 3*! = 3 × 2 × 1 = 6
• 0*! is defined as 1

Factorials appear in many areas of mathematics, especially in combinatorics, where they are used to count permutations and combinations. They also play a significant role in calculus, algebra, and number theory. Understanding how to manipulate and simplify expressions with factorials is crucial for solving various mathematical problems.

Breaking Down the Given Expression

Now, let's break down the given expression $rac{10!}{(10-3)!}$. First, we need to evaluate the terms in the expression. We have 10! in the numerator and (10-3)! in the denominator. Simplifying the denominator, we get:

(10 - 3)! = 7!

So, the expression becomes:

rac{10!}{7!}

Now, we need to expand the factorials to simplify the expression further. Expanding 10! and 7! gives us:

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Substituting these expansions into our expression, we have:

rac{10!}{7!} = rac{10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{7 × 6 × 5 × 4 × 3 × 2 × 1}

Simplifying the Expression

To simplify the fraction $rac{10!}{7!}$, we can cancel out the common terms in the numerator and the denominator. Notice that 7! appears in both the numerator and the denominator. Thus, we can cancel out the 7 × 6 × 5 × 4 × 3 × 2 × 1 terms:

rac{10!}{7!} = rac{10 × 9 × 8 × (7 × 6 × 5 × 4 × 3 × 2 × 1)}{(7 × 6 × 5 × 4 × 3 × 2 × 1)}

Cancelling out 7! from the numerator and the denominator, we are left with:

rac{10!}{7!} = 10 × 9 × 8

Now, we simply multiply the remaining numbers:

10 × 9 × 8 = 10 × 72 = 720

Therefore, the value of $rac{10!}{7!}$ is 720.

Conclusion

In conclusion, the value of the expression $rac{10!}{(10-3)!}$ is 720. We arrived at this answer by understanding the definition of factorials, expanding the factorials in the expression, and simplifying by canceling out common terms. This problem highlights the importance of understanding factorials and how to manipulate them, which is a common skill needed in many areas of mathematics, particularly in combinatorics and probability.

The final answer is 720, which corresponds to option C. This type of problem is fundamental in understanding permutations and combinations, which are essential concepts in discrete mathematics and are widely used in various fields such as computer science, statistics, and engineering. Mastering these concepts helps in problem-solving and analytical thinking, making it a valuable skill for students and professionals alike.

The correct option is C. 720

# Detailed Solution to Finding the Value of 10! / (10-3)!

## Introduction

The problem at hand involves the calculation of the value of the expression $rac{10!}{(10-3)!}$. This is a quintessential problem in mathematics that tests our understanding of factorials and their properties. Factorials are a fundamental concept in combinatorics, probability, and various other mathematical disciplines. They represent the product of all positive integers up to a given number. Solving this problem requires a clear understanding of what factorials mean and how to manipulate them effectively. This detailed explanation will guide you through the step-by-step process of solving this problem, ensuring clarity and comprehension of the underlying concepts.

## Understanding the Basics of Factorials

Before diving into the solution, it's crucial to grasp the definition of a factorial. The factorial of a non-negative integer *n*, denoted as *n*!, is the product of all positive integers less than or equal to *n*. Mathematically, this can be expressed as:

*n*! = *n* × (*n* - 1) × (*n* - 2) × ... × 3 × 2 × 1

For example, let's calculate a few factorials to illustrate this concept:

*   5*! = 5 × 4 × 3 × 2 × 1 = 120
*   4*! = 4 × 3 × 2 × 1 = 24
*   3*! = 3 × 2 × 1 = 6
*   2*! = 2 × 1 = 2
*   1*! = 1
*   0*! = 1 (by convention)

Factorials are extensively used in combinatorics to count the number of ways to arrange items (permutations) and to count the number of ways to choose items (combinations). They are also vital in calculus, especially in series expansions and differential equations. Understanding factorials forms a cornerstone for tackling problems in these diverse areas of mathematics.

## Breaking Down the Problem: $rac{10!}{(10-3)!}$

Now, let's return to the original problem: $rac{10!}{(10-3)!}$. The first step in solving this is to simplify the denominator. We have (10 - 3)!, which simplifies to 7!. So, the expression becomes:

$rac{10!}{7!}$

This expression involves the ratio of two factorials. To proceed, we need to expand the factorials. The factorial 10! means the product of all integers from 1 to 10, and 7! means the product of all integers from 1 to 7. Expanding these, we get:

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Substituting these expansions back into the expression, we have:

$rac{10!}{7!} = rac{10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{7 × 6 × 5 × 4 × 3 × 2 × 1}$

## Simplifying the Factorial Expression

To simplify the expression $rac{10!}{7!}$, we can observe that 7! is a common factor in both the numerator and the denominator. This allows us to cancel out the common terms. We can rewrite 10! as:

10! = 10 × 9 × 8 × (7 × 6 × 5 × 4 × 3 × 2 × 1) = 10 × 9 × 8 × 7!

Now, our expression looks like this:

$rac{10!}{7!} = rac{10 × 9 × 8 × 7!}{7!}$

By canceling out the 7! term in both the numerator and the denominator, we significantly simplify the expression:

$rac{10!}{7!} = 10 × 9 × 8$

This simplification makes the calculation much easier. We now only need to multiply three integers together.

## Performing the Multiplication

Now, let's perform the multiplication:

10 × 9 × 8 = 10 × (9 × 8) = 10 × 72 = 720

So, the value of the expression $rac{10!}{7!}$ is 720. This result demonstrates the power of simplifying factorial expressions by canceling out common factors, which greatly reduces the computational complexity.

## The Significance of Factorial Simplification

This problem underscores the importance of simplifying factorial expressions in mathematics. Factorials grow very quickly, and dealing with large factorials directly can be cumbersome and computationally intensive. By understanding how to cancel out common factors, especially in ratios of factorials, we can make complex calculations manageable. This skill is particularly useful in combinatorics, where factorial ratios appear frequently in formulas for permutations and combinations.

For instance, in calculating the number of ways to choose *k* items from a set of *n* items (combinations), we use the formula:

$C(n, k) = rac{n!}{k!(n-k)!}$

Simplifying these factorial expressions is essential to arrive at the correct answer efficiently. The ability to manipulate factorials is also crucial in calculus when dealing with Taylor series and other series expansions.

## Conclusion: The Value of $rac{10!}{(10-3)!}$

In conclusion, the value of the expression $rac{10!}{(10-3)!}$ is 720. We arrived at this result by first understanding the definition of factorials, expanding the factorials in the expression, simplifying by canceling out common terms, and then performing the multiplication. This problem is a foundational exercise in understanding factorial operations and showcases their practical application in various mathematical contexts. Mastery of such concepts is invaluable for anyone studying mathematics, computer science, engineering, or related fields, as it enhances problem-solving skills and analytical thinking.

Therefore, the final answer is 720, highlighting the importance of understanding and manipulating factorials in mathematical problem-solving.

# Finding the Value of 10! / (10-3)!: A Step-by-Step Guide

## Introduction to Factorials and the Problem

At the heart of many mathematical and computational problems lies the concept of **factorials**. Understanding ***factorials*** is essential for tackling questions in combinatorics, probability, and computer science. This article aims to provide a comprehensive, step-by-step solution to the problem of finding the value of the expression $rac{10!}{(10-3)!}$. We will delve into the definition of factorials, explore the simplification process, and arrive at the final answer. This problem serves as a fundamental exercise in manipulating factorials and underscores their importance in various mathematical contexts. ***Factorials*** are a cornerstone in understanding permutations, combinations, and other advanced mathematical concepts. This article provides a detailed walkthrough, ensuring that every step is clear and easy to follow.

## What is a Factorial?

Before we dive into the problem, let's define 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 mathematical terms, the ***factorial*** can be expressed as:

*n*! = *n* × (*n* - 1) × (*n* - 2) × ... × 3 × 2 × 1

For example, let's calculate a few ***factorials***:

*   5*! = 5 × 4 × 3 × 2 × 1 = 120
*   4*! = 4 × 3 × 2 × 1 = 24
*   3*! = 3 × 2 × 1 = 6
*   2*! = 2 × 1 = 2
*   1*! = 1
*   0*! = 1 (by definition)

The concept of ***factorials*** is not just an abstract mathematical idea; it has practical applications in various fields. ***Factorials*** are particularly important in combinatorics, where they are used to count the number of ways to arrange items (permutations) and to count the number of ways to choose items (combinations). They also appear in calculus, specifically in series expansions and differential equations. Thus, a strong grasp of ***factorials*** is essential for anyone studying mathematics, computer science, or related disciplines.

## Breaking Down the Problem: $rac{10!}{(10-3)!}$

Now that we have a solid understanding of **factorials**, let's tackle the problem at hand: $rac{10!}{(10-3)!}$. The first step is to simplify the expression within the parentheses in the denominator. We have (10 - 3)!, which simplifies to 7!. Therefore, the expression becomes:

$rac{10!}{7!}$

This expression involves a ratio of two **factorials**. To solve it, we need to understand how to expand and simplify ***factorial*** expressions. The next step involves expanding the ***factorials*** and looking for opportunities to simplify the expression. This is where the understanding of the basic properties of ***factorials*** becomes crucial.

## Expanding the Factorials

To proceed, we need to expand both 10! and 7!. Expanding a **factorial** means writing it out as the product of all positive integers up to that number. So, we have:

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Now, we substitute these expansions back into the expression:

$rac{10!}{7!} = rac{10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{7 × 6 × 5 × 4 × 3 × 2 × 1}$

At this stage, the expression looks quite large, but we're about to see how simplification can make it much more manageable. This is a critical skill in working with ***factorials***, especially when dealing with larger numbers. The ability to identify and cancel common factors is what makes these calculations feasible.

## Simplifying the Expression: Cancelling Common Factors

The key to simplifying expressions involving **factorials** lies in recognizing common factors. Looking at the expanded forms of 10! and 7!, we can see that 7! is a part of 10!. Specifically, 10! can be rewritten as:

10! = 10 × 9 × 8 × (7 × 6 × 5 × 4 × 3 × 2 × 1) = 10 × 9 × 8 × 7!

Now, our expression becomes:

$rac{10!}{7!} = rac{10 × 9 × 8 × 7!}{7!}$

We can now cancel out the 7! term from both the numerator and the denominator:

$rac{10!}{7!} = 10 × 9 × 8$

By canceling out the common **factorial**, we have greatly simplified the calculation. What started as a ratio of two large ***factorials*** is now a simple multiplication problem. This simplification technique is a fundamental aspect of working with ***factorials*** and is essential for solving more complex problems.

## Performing the Final Calculation

With the expression simplified, we can now perform the final multiplication:

10 × 9 × 8 = 10 × (9 × 8) = 10 × 72 = 720

So, the value of the expression $rac{10!}{7!}$ is 720. This result demonstrates the power of simplification techniques in mathematics. What initially seemed like a complex problem involving large ***factorials*** was reduced to a straightforward multiplication through strategic simplification.

## Conclusion: The Value of $rac{10!}{(10-3)!}$

In summary, we have found that the value of the expression $rac{10!}{(10-3)!}$ is 720. We achieved this by understanding the definition of **factorials**, expanding the ***factorials***, identifying and canceling common factors, and finally, performing the multiplication. This problem serves as an excellent example of how a strong understanding of basic mathematical concepts, combined with strategic problem-solving techniques, can lead to elegant solutions. ***Factorials*** are a recurring theme in various areas of mathematics, and mastering their manipulation is a valuable skill.

## The Importance of Understanding Factorials

This problem highlights the importance of understanding **factorials** and their properties. Factorials are not just mathematical curiosities; they have practical applications in various fields. In combinatorics, they are used to count permutations and combinations. In probability, they are used to calculate probabilities involving arrangements and selections. In computer science, they appear in algorithms related to sorting and searching. Therefore, a solid understanding of ***factorials*** is an asset for anyone pursuing studies or a career in these areas.

Moreover, the problem-solving techniques used here, such as expanding expressions and canceling common factors, are applicable to a wide range of mathematical problems. Learning to simplify complex expressions is a valuable skill that can be transferred to many different contexts. The ability to break down a problem into smaller, manageable steps is also crucial for effective problem-solving.

In conclusion, the journey from the initial problem to the final answer of 720 underscores the importance of understanding and manipulating **factorials**. This detailed walkthrough provides a clear and accessible explanation, ensuring that the underlying concepts are well-understood. By mastering these techniques, you will be well-equipped to tackle more complex problems involving ***factorials*** in the future.