Question

In Exercises \(\mathrm{P} .100\) to \(\mathrm{P} .107,\) calculate the requested quantity. $$ 6 ! $$

Short answer

The factorial of 6 is 720.

Step-by-step solution

Identify the number

The number given is 6, we are supposed to find the factorial of 6, denoted as 6!

Apply the factorial formula

Applying the factorial formula, we compute 6! = 6 x 5 x 4 x 3 x 2 x 1

Calculate the factorial

By multiplying these numbers together, we find that 6! = 720

Key concepts

Factorial Definition

In mathematics, the factorial of a non-negative integer is the product of all positive integers less than or equal to that integer. The factorial of a number is denoted by an exclamation point (!). For instance, when we say 4!, we're referring to the factorial of 4, which involves multiplying all the positive integers from 4 down to 1.

Factorials are widely used in subjects like algebra, calculus, and especially in combinations and permutations within probability and statistics to calculate the number of ways things can be arranged or combined. The factorial of zero is defined as 1, which is a convention in mathematics for convenience in various formulas. This definition guides us when dealing with n!, where n is the number of interest.

Special note: Factorials are only defined for non-negative integers. This is because the concept relies on a finite, descending list of numbers to multiply, which does not exist for fractions, irrationals, or negative numbers.

Factorial Formula

The factorial formula is quite straightforward. For any non-negative integer n, its factorial, denoted as n!, is calculated as:
\[ n! = n \times (n - 1) \times (n - 2) \times \dots \times 3 \times 2 \times 1 \]
The product starts at the given number and decreases by one until it reaches 1. A fundamental point to remember is that the factorial of 0 is always 1, denoted as \( 0! = 1 \). This might seem odd at first, but it's important to understand that this is a defined value that ensures the consistency of mathematical equations and concepts such as the binomial theorem and combinations.

Let's illustrate this with a practical example: 4! (the factorial of 4) would be computed as \( 4 \times 3 \times 2 \times 1 = 24 \).
Regularly using the factorial formula is critical when computing combinations and permutations.

Computing Factorials

The process of computing factorials can be done manually or with the aid of calculators or computer software, especially for larger numbers which can result in very large products. Here are a few tips for computing factorials:

• Start with the highest number and work downwards.
• Remember that multiplying by 1 does not change the value, so you can stop the multiplication once you reach 2.
• For large numbers, use calculators or software to prevent errors and save time.

Using the example given in the original exercise, to calculate 6!, you would apply the factorial formula as such: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] which equals 720 after performing the multiplication.

Such calculations become fundamental when dealing with more complex mathematical topics, such as evaluating the number of permutations of objects, where knowing how to efficiently compute factorials is invaluable.