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Magazine Chapter 5: Problem 7
Find the value of each factorial. \(10 !\)
Short Answer
Expert verified
10! = 3628800
Step by step solution
01
Understand the concept of a factorial
A factorial, denoted by an exclamation mark after a number, is the product of all positive integers from 1 up to that number. For example, the factorial of 4, written as 4!, is calculated as 4 脳 3 脳 2 脳 1.
02
Write down the factorial expression
For the given exercise, write down the number and its factorial expression: 10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
03
Perform the multiplication
Calculate the product step-by-step:10 脳 9 = 9090 脳 8 = 720720 脳 7 = 50405040 脳 6 = 3024030240 脳 5 = 151200151200 脳 4 = 604800604800 脳 3 = 18144001814400 脳 2 = 36288003628800 脳 1 = 3628800
04
Verify your calculation
Verify your result by reconsidering and reperforming each multiplication step to ensure accuracy.After performing all checks, conclude: 10! = 3628800
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
mathematical expression
In mathematics, a mathematical expression is a combination of numbers, symbols, and operators that represent a value or relationship. Understanding these expressions is crucial to solving problems.
For instance, in the exercise given, we are asked to calculate the factorial of 10, denoted as 10!. This expression tells us to multiply a sequence of descending natural numbers. Here, understanding the notation and how to expand it into a mathematical expression is the first step.
The expression for 10! is expanded as follows:
10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
Such clarity in expressing the sequence is essential for further calculations. Therefore, mastering the identification and conversion of mathematical expressions into solvable steps is fundamental.
For instance, in the exercise given, we are asked to calculate the factorial of 10, denoted as 10!. This expression tells us to multiply a sequence of descending natural numbers. Here, understanding the notation and how to expand it into a mathematical expression is the first step.
The expression for 10! is expanded as follows:
10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
Such clarity in expressing the sequence is essential for further calculations. Therefore, mastering the identification and conversion of mathematical expressions into solvable steps is fundamental.
factorial calculation
A factorial calculation involves multiplying a series of descending natural numbers. The factorial of a number n, symbolized as n!, means multiplying all positive integers up to n.
For example, 4! (read as 'four factorial') is calculated as:
4! = 4 脳 3 脳 2 脳 1 = 24.
In our exercise, we need to find 10!, which means we multiply these numbers:
10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
Let's break it down:
Through an organized approach to each step, we ensure accuracy and completion of the factorial calculation. Verifying each multiplication helps avoid potential mistakes. By breaking down the factorial computation into smaller steps, the overall process becomes easier to follow and comprehend.
For example, 4! (read as 'four factorial') is calculated as:
4! = 4 脳 3 脳 2 脳 1 = 24.
In our exercise, we need to find 10!, which means we multiply these numbers:
10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
Let's break it down:
- First, calculate 10 脳 9 = 90.
- Then, multiply the result by 8: 90 脳 8 = 720.
- Next, 720 脳 7 = 5040.
- Continuing, 5040 脳 6 = 30240;
- then 30240 脳 5 = 151200;
- 151200 脳 4 = 604800;
- 604800 脳 3 = 1814400;
- 1814400 脳 2 = 3628800;
- and finally 3628800 脳 1 = 3628800.
Through an organized approach to each step, we ensure accuracy and completion of the factorial calculation. Verifying each multiplication helps avoid potential mistakes. By breaking down the factorial computation into smaller steps, the overall process becomes easier to follow and comprehend.
multiplication
Multiplication, one of the basic arithmetic operations, is a process of adding a number to itself a certain number of times. Understanding multiplication is essential in performing and verifying factorial calculations.
For instance, consider the sequence of multiplications needed to find 10!. Performing each multiplication step-by-step ensures clarity:
Each multiplication builds on the previous one, adding complexity and magnitude to the calculations. Therefore, ensuring the accuracy of each multiplication step is crucial, especially when dealing with large numbers like in factorials.
Checking and rechecking each multiplication step helps avoid errors and confirms the final result. It's helpful to write down interim results like shown above to track and verify each multiplication stage easily.
For instance, consider the sequence of multiplications needed to find 10!. Performing each multiplication step-by-step ensures clarity:
- 10 脳 9 = 90
- 90 脳 8 = 720
- 720 脳 7 = 5040
- 5040 脳 6 = 30240
- 30240 脳 5 = 151200
- 151200 脳 4 = 604800
- 604800 脳 3 = 1814400
- 1814400 脳 2 = 3628800
- 3628800 脳 1 = 3628800
Each multiplication builds on the previous one, adding complexity and magnitude to the calculations. Therefore, ensuring the accuracy of each multiplication step is crucial, especially when dealing with large numbers like in factorials.
Checking and rechecking each multiplication step helps avoid errors and confirms the final result. It's helpful to write down interim results like shown above to track and verify each multiplication stage easily.
