Evaluate The Expression:$\frac{9!}{3!}$A. 3 B. 6 C. 60,480 D. 362,874

Introduction

Factorials are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on evaluating the expression $\frac{9!}{3!}$, which involves calculating the factorial of 9 and dividing it by the factorial of 3.

What are 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. In other words, it is the result of multiplying all whole numbers from n down to 1. For example, the factorial of 5 (denoted by 5!) is equal to 5 × 4 × 3 × 2 × 1 = 120.

Evaluating the Expression

To evaluate the expression $\frac{9!}{3!}$, we need to calculate the factorial of 9 and divide it by the factorial of 3.

Calculating 9!

To calculate 9!, we need to multiply all whole numbers from 9 down to 1.

9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880

Calculating 3!

To calculate 3!, we need to multiply all whole numbers from 3 down to 1.

3! = 3 × 2 × 1 = 6

Evaluating the Expression

Now that we have calculated 9! and 3!, we can evaluate the expression $\frac{9!}{3!}$ by dividing 9! by 3!.

$\frac{9!}{3!} = \frac{362,880}{6}$ = 60,480

Conclusion

In this article, we evaluated the expression $\frac{9!}{3!}$ by calculating the factorial of 9 and dividing it by the factorial of 3. We found that the result of the expression is 60,480. Understanding how to evaluate factorials is crucial for solving various mathematical problems, and this article provides a step-by-step guide on how to do it.

Common Mistakes to Avoid

When evaluating factorials, it's essential to avoid common mistakes such as:

• Not calculating the factorial correctly: Make sure to multiply all whole numbers from n down to 1 to get the correct result.
• Not dividing the factorials correctly: When dividing factorials, make sure to divide the larger factorial by the smaller one.
• Not simplifying the expression: Simplify the expression by canceling out common factors.

Real-World Applications

Understanding how to evaluate factorials has numerous real-world applications, including:

• Computer Science: Factorials are used in computer science to calculate permutations and combinations.
• Data Analysis: Factorials are used in data analysis to calculate probabilities and statistics.
• Engineering: Factorials are used in engineering to calculate stress and strain on materials.

Practice Problems

To practice evaluating factorials, try the following problems:

• Evaluate the expression $\frac{5!}{2!}$.
• Evaluate the expression $\frac{7!}{4!}$.
• Evaluate the expression $\frac{10!}{6!}$.

Conclusion

Introduction

In our previous article, we discussed how to evaluate the expression $\frac{9!}{3!}$, which involves calculating the factorial of 9 and dividing it by the factorial of 3. In this article, we will provide a Q&A guide to help you understand the concept of factorials and how to evaluate them.

Q: What is a factorial?

A: A 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, it is the result of multiplying all whole numbers from n down to 1.

Q: How do I calculate a factorial?

A: To calculate a factorial, you need to multiply all whole numbers from n down to 1. For example, to calculate 5!, you would multiply 5 × 4 × 3 × 2 × 1 = 120.

Q: What is the difference between 5! and 5?

A: 5! is the product of all whole numbers from 5 down to 1, which is 5 × 4 × 3 × 2 × 1 = 120. On the other hand, 5 is just the number 5 itself.

Q: How do I evaluate an expression with factorials?

A: To evaluate an expression with factorials, you need to calculate the factorial of each number and then perform the operations as indicated in the expression. For example, to evaluate the expression $\frac{9!}{3!}$, you would calculate 9! and 3! and then divide 9! by 3!.

Q: What is the result of $\frac{9!}{3!}$?

A: The result of $\frac{9!}{3!}$ is 60,480.

Q: Can I use a calculator to evaluate factorials?

A: Yes, you can use a calculator to evaluate factorials. However, it's essential to understand the concept of factorials and how to calculate them manually to ensure accuracy.

Q: How do I simplify an expression with factorials?

A: To simplify an expression with factorials, you can cancel out common factors. For example, to simplify the expression $\frac{9!}{3!}$, you can cancel out the 3! in the numerator and denominator, leaving you with $\frac{9 \times 8 \times 7 \times 6 \times 5 \times 4}{1}$.

Q: What are some real-world applications of factorials?

A: Factorials have numerous real-world applications, including:

• Computer Science: Factorials are used in computer science to calculate permutations and combinations.
• Data Analysis: Factorials are used in data analysis to calculate probabilities and statistics.
• Engineering: Factorials are used in engineering to calculate stress and strain on materials.

Q: How do I practice evaluating factorials?

A: You can practice evaluating factorials by trying the following problems:

• Evaluate the expression $\frac{5!}{2!}$.
• Evaluate the expression $\frac{7!}{4!}$.
• Evaluate the expression $\frac{10!}{6!}$.

Conclusion

In conclusion, evaluating factorials is a crucial concept in mathematics that has numerous real-world applications. By understanding how to calculate factorials and divide them correctly, you can solve various mathematical problems and apply the concepts to real-world scenarios. We hope this Q&A guide has helped you understand the concept of factorials and how to evaluate them.