When we apply the Multiplication Principle and do not allow repetition, the number of choices in each part of the product drops by \(1 .\) This leads to products like \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) or \(8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)
This type of product occurs so often that it is assigned its own symbol.
Factorial Notation
For any positive integer \(n\)
\[
n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1
\]
The value of \(0 !\) is defined to be 1 .
Let's look at how we might apply this to an application.
Suppose a production line requires six workers to carry out six different jobs. Each worker can only do one job at a time. Once a worker is selected for a job, the other jobs must be carried out by the remaining workers. To find the number of ways we can assign workers to jobs, calculate the product
The number of ways to make each choice drops by one in each factor since each worker can only do one job. In effect, we can't choose the same worker twice. This is often indicated by saying that we want to assign workers without repetition.
