Evaluate The Expression:$4! \cdot 3!$A. 12 B. 30 C. 144 D. 5040

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Understanding Factorials

In mathematics, a factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. Factorials are used to calculate the number of permutations of a set of objects.

Calculating 4! and 3!

To evaluate the expression 4! * 3!, we need to calculate the factorials of 4 and 3.

  • 4! = 4 * 3 * 2 * 1 = 24
  • 3! = 3 * 2 * 1 = 6

Multiplying 4! and 3!

Now that we have the values of 4! and 3!, we can multiply them together to get the final result.

  • 4! * 3! = 24 * 6 = 144

Conclusion

Therefore, the correct answer is C. 144.

Understanding the Importance of Factorials

Factorials are used in various mathematical operations, such as calculating the number of permutations of a set of objects. They are also used in probability theory, statistics, and number theory.

Real-World Applications of Factorials

Factorials have numerous real-world applications, including:

  • Computer Science: Factorials are used in algorithms for calculating permutations and combinations.
  • Data Analysis: Factorials are used in statistical analysis to calculate the number of possible outcomes in a given scenario.
  • Cryptography: Factorials are used in cryptographic algorithms to calculate the number of possible keys.

Common Mistakes in Calculating Factorials

When calculating factorials, it's essential to remember the following:

  • Order of Operations: Factorials should be calculated in the correct order, starting from the innermost parentheses.
  • Negative Numbers: Factorials are not defined for negative numbers.
  • Non-Integer Values: Factorials are not defined for non-integer values.

Tips for Calculating Factorials

To calculate factorials efficiently, follow these tips:

  • Use a Calculator: Use a calculator to calculate large factorials.
  • Break Down the Problem: Break down the problem into smaller sub-problems to make it easier to calculate.
  • Use a Formula: Use a formula to calculate factorials, such as n! = n * (n-1) * (n-2) * ... * 1.

Conclusion

In conclusion, factorials are a fundamental concept in mathematics, and understanding how to calculate them is essential for various mathematical operations. By following the tips and avoiding common mistakes, you can calculate factorials efficiently and accurately.

Final Answer

The final answer is C. 144.

Q&A: Evaluating the Expression 4! * 3!

Q: What is the value of 4! * 3!?

A: The value of 4! * 3! is 144.

Q: How do I calculate 4! and 3!?

A: To calculate 4! and 3!, you need to multiply the numbers together in the correct order. For example:

  • 4! = 4 * 3 * 2 * 1 = 24
  • 3! = 3 * 2 * 1 = 6

Q: What is the formula for calculating factorials?

A: The formula for calculating factorials is:

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

Q: Can I use a calculator to calculate factorials?

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

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

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

  • Computer Science: Factorials are used in algorithms for calculating permutations and combinations.
  • Data Analysis: Factorials are used in statistical analysis to calculate the number of possible outcomes in a given scenario.
  • Cryptography: Factorials are used in cryptographic algorithms to calculate the number of possible keys.

Q: What are some common mistakes to avoid when calculating factorials?

A: Some common mistakes to avoid when calculating factorials include:

  • Order of Operations: Factorials should be calculated in the correct order, starting from the innermost parentheses.
  • Negative Numbers: Factorials are not defined for negative numbers.
  • Non-Integer Values: Factorials are not defined for non-integer values.

Q: How can I calculate factorials efficiently?

A: To calculate factorials efficiently, follow these tips:

  • Use a Calculator: Use a calculator to calculate large factorials.
  • Break Down the Problem: Break down the problem into smaller sub-problems to make it easier to calculate.
  • Use a Formula: Use a formula to calculate factorials, such as n! = n * (n-1) * (n-2) * ... * 1.

Q: What is the importance of understanding factorials?

A: Understanding factorials is essential for various mathematical operations, including probability theory, statistics, and number theory. It's also crucial for real-world applications, such as computer science, data analysis, and cryptography.

Conclusion

In conclusion, evaluating the expression 4! * 3! requires understanding the concept of factorials and how to calculate them. By following the tips and avoiding common mistakes, you can calculate factorials efficiently and accurately.

Final Answer

The final answer is C. 144.

Additional Resources

  • Factorial Formula: n! = n * (n-1) * (n-2) * ... * 1
  • Real-World Applications: Computer Science, Data Analysis, Cryptography
  • Common Mistakes: Order of Operations, Negative Numbers, Non-Integer Values
  • Efficient Calculation: Use a Calculator, Break Down the Problem, Use a Formula