What is 1 factorial ?

The factorial of a number, typically represented as n!, is the product of all positive integers less than or equal to n. When we talk about the factorial of 1, we’re focusing on a fundamental aspect of factorials, as it forms the basis for understanding this mathematical operation. The factorial of one is significant because it defines the starting point for the factorial function and provides a cornerstone for mathematical proofs and applications.

The general formula to calculate the factorial of a number is given as n! = n × (n-1) × … × 1. For the number 1, the factorial can be calculated by applying this formula:

• Start with the number itself: 1
• Since there are no other numbers below 1, the product ends with the number itself

Therefore, 1! = 1 × (1) = 1, which illustrates that the factorial of 1 is simply 1.

The factorial function, including the factorial of 1, has several practical applications in fields like mathematics and science. Here are a few:

• Combinatorics: The factorial of 1 comes into play when determining the number of ways to arrange a single item, which is always one way.
• Probability Theory: When calculating probabilities and permutations where you have only one item or event, the factorial of 1 ensures the equations remain consistent and valid.
• Algebra: Factorials, including 1!, are used in expansions and expressions within algebraic equations.

1. Calculate the value of 0! and explain how it relates to 1!.
2. If you have 1 book, in how many ways can you arrange it on a shelf?
3. Write the equation for (1 + 1)! and calculate its value.

1. The value of 0! is 1, as there is only one way to arrange nothing. This is the foundation for defining 1! as 1, since both represent the number of arrangements of 0 and 1 item respectively.
2. You can arrange 1 book on a shelf in exactly 1 way, hence the answer is 1! which equals 1.
3. (1 + 1)! is the factorial of 2, so the equation is 2! = 2 × 1, and its value is 2.