In Try This 3 you found that you could calculate the number of ways ten children could be lined up by finding the product of all consecutive natural numbers less than or equal to ten.

In mathematics there is a short-cut notation for the product of a positive integer, n, and all integers less than or equal to n. This short cut is called factorial notation and is denoted n!. Using factorial notation, the solution to Try This 3 would be that students can be lined up in 10! ways.

To learn more about how factorial notation works, work through Number of Arrangements. You will determine how many ways ten people can be arranged in the lineup. To begin, you will be asked to state how many people are available to fill the first spot in line.

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© Peter Vaclavek
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Self-Check 3

1. What is the value of 5!? Answer
   
2. How can 6 × 5 × 4 × 3 × 2 × 1 be written using factorial notation? Answer

You have seen that 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

If you let n = 10, then the expression can be written as the following:

n! = n × (n − 1) × (n − 2) × (n − 3) × (n − 4 ) × ( n − 5) × ( n − 6) × ( n − 7) × ( n − 8) × ( n − 9)

or

n! = n(n − 1)(n − 2)(n − 3)…(3)(2)(1)

Calculating the value of an expression involving factorial notation without a calculator can be time consuming. In Try This 4 you will look for patterns that might exist between subsequent expressions involving factorial notation.

Try This 4

1. Complete the following table based on the information already available.
   
   

   n!	n! in Expanded Form	Expression Involving n and (n − 1)!	Value of n!
   1!	1	1	1
   2!	2 × 1	2 × _!	2
   3!	3 × 2 × 1	3 × _!
   4!	_ × _ × _ × _	4 × _!
   5!	_ × _ × _ × _ × _	5 × _!
   6!	_ × _ × _ × _ × _ × _	6 × _!
   7!	_ × _ ×  _ × _ × _ × _ × _	7 × _!

2. Describe the relationship between n! and (n − 1)!.

3. Create a general expression relating n!, n, and (n − 1)!.

Save your responses in your course folder.