The concept and notation of factorials are identified.
After completing this tutorial, you will be able to complete the following:
An examination of the counting principle leads to an explanation of why factorial notation is important. The counting principle is a method for calculating all of the possible outcomes of one event or possible arrangement of an object with another event or object. You multiply the number of possibilities.
For example, suppose you have three pictures to hang horizontally on a wall. You can lay the pictures down, as shown here, to see how many arrangements you have.
That is quite time consuming. A shorter way is to multiply to find all the possible combinations of how the pictures can be hung:
3 X 2 X 1 = 6
In general, the number of ways n objects can be arranged is the product of all positive integers less than or equal to n.
This is fine with smaller numbers of objects, but multiplication becomes cumbersome with large numbers. For example, if you have 10 pictures to hang in 10 spots on the wall, how many possible arrangements is that? The first picture can be put in 10 spots. With that picture in place, the second picture can be put in 9 spots, and so on. We find:
10 X 9 X 8 X ... X 1 = 3,628,800
This illustrates the need for the use of factorials.
A factorial is the product of all positive integers less than or equal to n. The number of sequential arrangements for n objects is "n factorial," written as n!
10! = 10 X 9 X 8 X ... X 1 = 3,628,800
The following expression is helpful in finding factorials:
n! = n (n 1)!
It is derived from the counting principle as shown here:
The expression n! = n × (n - 1)! helps in many calculations involving factorial notation.
If you are arranging no objects, you still have a set. This is an empty set, and the number of sets is 1. The proof for 0! follows:
The number of arrangements of 0 objects = 1. When multiplying by 0!, the answer doesn't change, as multiplying by 0! is the same as multiplying by 1. In mathematics the product of multiplying by no numbers is 1.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know how to use the counting principle, division, and multiplication, and understand the concept of positive integers.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||arrangement, fundamental counting principle, counting techniques|