Factorial notation, denoted by |n!|, is a way to write the product of all the positive integers less than or equal to a number |n|, where |n| is a natural number.
This notation also simplifies the calculations and the process for solving.
||n! = n\times (n-1)\times (n-2)\times \dots \times 3\times 2\times 1||
Factorial notation makes it possible to simplify the writing of the mathematical operation. Instead of writing the product of all the integers involved, it is only necessary to write the integer whose factorial is to be calculated, followed by an exclamation point.
Example 1
||\begin{align} 3! &= 3 \times 2 \times 1 \\
&= 6 \end{align}||
Example 2
||\begin{align} 9! &= 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
&= 362 \ 880\end{align}||
Example 3
||\begin{align} 13! &= 13 \times 12 \times ... \times 3 \times 2 \times 1\\
&= 6 \ 227 \ 020 \ 800 \end{align}||
As with many mathematical operations, |0| is a special case.
By convention, |0!=1|.
As for the practical use of factorial notation, it’s often used in probability to determine the number of possible permutations of the elements of a set.
Question
A draw determines the order in which the |5| performances of a show will be presented. How many possibilities are there?
Answer
This situation involves |5| events. So there are |5| possible choices for the first number. After this, this means that there are only |4| possible choices for the second number, and so on. To determine the total number of possibilities for the sequence of numbers, simply multiply everything together. ||\begin{align} \text{Total number of possibilities} &= 5\times 4\times 3\times 2\times 1 \\ &= 5! \\ &= 120\end{align}||