Factorials, represented by the symbol (!), are essential mathematical functions that have various uses such as finding permutations, combinations, and arrangements. Let's delve into the basics of factorials, how to calculate them, and their practical applications.

The formula for factorials is n!, where n is any whole number, and the exclamation mark denotes the expression as factorial. Now, let's take a closer look at the steps for calculating factorials and some examples of their use.

To calculate the factorial of a number n, follow these steps:

- Write down the sequence of numbers you will multiply, using the factorial formula.
- Substitute the number n in for each instance of n in the formula until the final subtraction is equal to n-(n-1).
- Perform the necessary operations on the numbers you have written down, subtracting and multiplying where needed.

For example, let's find the factorial of 6, expressed as 6!. Following these steps, we have 6 × 5 × 4 × 3 × 2 × 1, which equals 720.

Factorials have various practical uses, such as determining possible combinations in a particular order. For instance, let's say we have the colors blue, red, and yellow, and we want to know the number of combinations that can be formed with them in a specific order. In this case, we can use factorials to find the answer. 3! = 3 × 2 × 1 = 6. The possible combinations are:

- Blue, red, and yellow
- Blue, yellow, and red
- Yellow, blue, and red
- Yellow, red, and blue
- Red, blue, and yellow
- Red, yellow, and blue

Another example could be determining the number of ways the letters in a word can be arranged without repeating. In this case, we first count the number of letters, let's say 7, and then find its factorial. The maximum number of ways is 7! = 5040, with no repeats.

Let's explore some examples of operations involving factorials, such as addition, subtraction, multiplication, and division. Note that when multiplying two factorials, the notation is (n!) (m!), not n! m!.

To add two factorials, simply add the numbers inside the parentheses. For example: 3! + 2! = 6.

When subtracting factorials, subtract the numbers inside the parentheses. For example: 5! - 3! = 120 - 6 = 114.

Multiplying two factorials involves multiplying the numbers inside the parentheses. For example: 3! × 4! = 6 × 24 = 144.

To divide one factorial by another, divide the numbers inside the parentheses. For example: 6! ÷ 3! = 720 ÷ 6 = 120.

Sometimes we encounter situations with variables and factorials. In such cases, we can use the factorial rule to simplify expressions. For instance, let's take the expression (n-1)!. When n = 7, this simplifies to 7!. By canceling out the common term (n-1)!, we get n. Similarly, when n = 10, (n-1)! also cancels out, leaving us with n+1.

- Factorials are mathematical functions represented by the symbol (!) that have various practical uses, such as determining permutations, combinations, and arrangements.
- To calculate a factorial, multiply a number by every number below it. For example, 4! = 4 × 3 × 2 × 1 = 24.
- Factorials can be used in various situations, including finding possible combinations in a given order.
- The rule for factorials is to multiply all the whole numbers from the chosen number down to one, expressed as n! = n × (n-1)!
- For factorials with variables, we can simplify expressions algebraically using the factorial rule.

Now that you have a good understanding of factorials, let's practice with a problem. Evaluate the expression below:

3! × (4! ÷ 2!)

Answer: 432

To solve this, first find the value of each factorial separately. 3! = 6 and 4! = 24. Then, divide 24 by 2!, resulting in 12. Finally, multiply 6 and 12 to get 72.

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