Factorials are a mathematical operation represented by an exclamation point (!), indicating the product of all whole numbers from a given number down to one. For instance, the factorial of four, denoted as 4!, is calculated as:
4! = 4 × 3 × 2 × 1 = 24
Factorials are essential in various fields such as sequences, series, combinatorics, and probability. The factorial of one, 1!, is simply one since there are no numbers to multiply down to. Similarly, the factorial of two, 2!, is:
2! = 2 × 1 = 2
For three, 3! is:
3! = 3 × 2 × 1 = 6
Continuing this pattern, we find:
4! = 24
5! = 5 × 4! = 5 × 24 = 120
6! = 6 × 5! = 6 × 120 = 720
In general, for any integer n, the factorial can be expressed as:
n! = n × (n - 1)!
This recursive relationship allows for simplification of factorial expressions. For example, 4 × 3! can be simplified to:
4 × 3! = 4! = 24
Another example is 100! / 99!. Instead of calculating both factorials fully, we can rewrite it as:
100! = 100 × 99!
Thus, the expression simplifies to:
100! / 99! = 100
Additionally, the concept of zero factorial, 0!, is defined as one. This can be derived from the relationship:
1! = 1 × 0!
Since 1! = 1, it follows that:
1 = 1 × 0!
Therefore, 0! = 1.
Understanding factorials and their properties is crucial for solving problems in mathematics, particularly in combinatorial contexts. Practice with various factorial expressions will enhance your proficiency in this area.
