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In Mathematics, factorial is a simple thing. Factorials are just products. An exclamation mark indicates the factorial. Factorial is a multiplication operation of natural numbers with all the natural numbers that are less than it. In this article, let’s discuss the factorial definition, formula and examples.
let’s calculate the factorial of 10 step by step:
Step 1: Start with the number whose factorial we want, which is 10. Step 2: Write down all the positive integers less than or equal to 10: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1. Step 3: Multiply these numbers together:
10!=10×9×8×7×6×5×4×3×2×110!=10×9×8×7×6×5×4×3×2×1
Now, let’s calculate the factorial:
10!=10×9×8×7×6×5×4×3×2×110!=10×9×8×7×6×5×4×3×2×1 10!=10×9×8×7×6×5×4×3×2×110!=10×9×8×7×6×5×4×3×2×1 10!=10×36288010!=10×362880 10!=362880010!=3628800
So, the factorial of 10 (10!) is 3628800.
Let’s break down the calculation of the factorial of 100 step by step:
Step 1: Start with the number whose factorial we want, which is 100. Step 2: Write down all the positive integers less than or equal to 100: 100, 99, 98, …, 3, 2, 1. Step 3: Multiply these numbers together:
100! = 100 × 99 × 98 × … × 3 × 2 × 1
Now, calculating this manually would be quite arduous, but we can utilize the scientific notation to represent this large number more conveniently.
Using the provided formula for factorials, we multiply each number together from 100 down to 1. This results in a massive number, represented in scientific notation as 9.332621544 × 10^157.
So, the factorial of 100 (100!) is approximately 9.332621544 × 10^157.
By breaking down the calculation and utilizing scientific notation, we can better understand the magnitude of the factorial of 100. It’s a truly enormous number, showcasing the power of factorials and the importance of proper steps in calculating them.
let’s break down the calculation for the factorial of 1000 in a similar step-by-step manner:
Step 1: Start with the number whose factorial we want, which is 1000. Step 2: Write down all the positive integers less than or equal to 1000: 1000, 999, 998, …, 3, 2, 1. Step 3: Multiply these numbers together:
1000!=1000×999×998×…×3×2×11000!=1000×999×998×…×3×2×1
Now, performing the multiplication:
1000!=1000×999×998×…×3×2×11000!=1000×999×998×…×3×2×1
The factorial of 1000 is the product of all positive integers less than or equal to 1000.
Let’s calculate the numerical result:
1000!=1000×999×998×…×3×2×11000!=1000×999×998×…×3×2×1
1000!=1000×999×998×…×3×2×11000!=1000×999×998×…×3×2×1
1000!=a very large number1000!= 4.023 x 10^2567
Think of factorials as a mathematical journey where each number leads to its factorial destination! To calculate the factorial of a number, you’re essentially summoning all the smaller factorial values to join forces in a grand multiplication parade.
Imagine starting with the number you want the factorial of, let’s call it ‘n’. You beckon all the positive integers smaller than or equal to ‘n’ to line up and contribute to the factorial mission. They obediently line up, from ‘n’ all the way down to 1.
Then, you unleash the mighty power of multiplication, each number in the lineup taking its turn to join forces with the others. It’s like a factorial orchestra, with each number playing its crucial part.
And guess what? Even when ‘n’ is 0, it’s not left out! It bravely stands at the front of the line, and in a magical mathematical twist, its factorial value is declared as 1! It’s like the mathematical equivalent of saying, “Even zero has its own unique power.”
So, the factorial of ‘n’, denoted as ‘n!’, is the result of this epic multiplication journey, where every number plays its part, and even the seemingly empty lineup of zero has a value of 1. It’s a mathematical symphony of multiplication where every number finds its factorial destiny!
Step 1: Start with the number whose factorial we want, which is 5. Step 2: Write down all the positive integers less than or equal to 5: 5, 4, 3, 2, 1. Step 3: Multiply these numbers together: 5 × 4 × 3 × 2 × 1 = 120.
So, the factorial of 5 (5!) is 120.
Let’s visualize it step by step:
Step 1: Start with 5! = 5 Step 2: Multiply by 4! = 4 × 3 × 2 × 1 = 24 Step 3: Multiply by 3! = 3 × 2 × 1 = 6 Step 4: Multiply by 2! = 2 × 1 = 2 Step 5: Multiply by 1! = 1
Now, let’s multiply the results of each step together: 5! = 5 × 4! = 5 × 4 × 3! = 5 × 4 × 3 × 2! = 5 × 4 × 3 × 2 × 1! = 5 × 24 = 120
So, the factorial of 5 (5!) is 120, and each step shows how we arrive at that result by progressively multiplying by the factorial of the previous number.
Here, the table of factorials has been given for numbers from 1 to 20.
n | n! |
1 | 1 |
2 | 2 |
3 | 6 |
4 | 24 |
5 | 120 |
6 | 720 |
7 | 5040 |
8 | 40320 |
9 | 362880 |
10 | 3628800 |
11 | 39916800 |
12 | 479001600 |
13 | 6227020800 |
14 | 87178291200 |
15 | 1307674368000 |
16 | 20922789888000 |
17 | 355687428096000 |
18 | 6402373705728000 |
19 | 121645100408832000 |
20 | 2432902008176640000 |
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