Factorial Calculator (n!)

Instructions:

• Enter a non-negative integer in the input field.
• Click "Calculate Factorial" to calculate the factorial.
• View the result, detailed calculation, and formula.
• Your calculation history will be displayed below.
• Click "Clear Results" to reset the results and history.
• Click "Copy Results" to copy the result and explanation to the clipboard.

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Factorials, denoted by n!, represent the product of all positive integers from 1 to n. This mathematical concept finds applications in various fields, including combinatorics, probability, and statistics. Understanding factorials and their properties is essential for solving a wide range of mathematical problems.

The Essence of Factorials: Terminology and Formula

Factorial: The factorial of a non-negative integer n is the product of all positive integers from 1 to n. It is denoted by n!, where n is a non-negative integer.

Formula: The factorial of a non-negative integer n can be calculated using the following formula:

n! = 1 * 2 * 3 * ... * n

where n is a non-negative integer.

Special Cases:

• 0! = 1 (by convention)
• 1! = 1

Navigating the Laws of Factorials: Properties and Simplification Rules

Factorials adhere to specific rules that govern their manipulation and simplification. These properties are essential for solving mathematical problems involving factorials.

Product of Factorials with the Same Base:

a^m * a^n = a^(m + n)

Power of a Factorial:

(a^m)^n = a^(m * n)

Factorial of a Product:

(a * b)^n = a^n * b^n

Quotient of Factorials with the Same Base:

a^m / a^n = a^(m - n)

Benefits of Factorials: Applications and Advantages

Factorials offer numerous benefits and advantages in various mathematical and scientific fields:

• Combinatorics: Factorials are crucial in combinatorics, the study of arrangements and combinations of objects. They are used to calculate the number of ways to arrange or select objects from a larger set.
• Probability: Factorials are fundamental in probability theory, particularly in discrete probability distributions. They are used to calculate the probability of specific events occurring.
• Statistics: Factorials are employed in statistical analysis, particularly in hypothesis testing and statistical inference. They are used to calculate p-values and confidence intervals.

Intriguing Facts and Applications of Factorials

• Factorials grow rapidly with increasing n values. For instance, 10! is approximately 3.6288 x 10^6.
• Factorials are used to approximate the number of permutations and combinations of large sets of objects.
• Factorials are used in algorithms for generating random numbers and shuffling data structures.

References

• “Concrete Mathematics” by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (1994)
• “Introduction to Probability” by Joseph K. Blitzstein and Jessica Hwang (2014)
• “Combinatorics and Probability” by Graham R. Brightwell and Timothy J. Ott (2009)