1. 2.1 Factorial Notation (Textbook Section 4.6)

2. Warm – Up Question  How many four-digit numbers can be made using the numbers 1, 2, 3, & 4?  (all numbers must only be used once for each 4-digit number)

3. Fundamental Principle of Counting  If one operation can be done in “m” ways and another operation can be done in “n” ways, then together, they can be done in mxn ways  This principle can be extended to any number of operations  i.e. if operation A can be done 3 ways, operation B can be done 4 ways, operation C can be done in 2 ways and operation D can be done 7 ways, then together they can be done in 3 x 4 x 2 x 7 = 168 ways

4. Back to Warm Up Question  How many ways can we place the number 1?  1 can be the 1 st, 2 nd, 3 rd or 4 th digit, so 4 ways  IF we have placed the number 1, how many ways can we place the number 2?  One of the digits has already been taken up by the number 1, so there are 3 remaining digit places to put the number 2, so 3 ways

5. Back to Warm Up Question (Continued)  IF we have placed numbers 1 and 2, how many ways can we place the number 3?  2 remaining digit places, so 2 ways  IF we have placed numbers 1, 2, and 3, how many ways can we place the number 4?  One spot remaining, place 1 there, so 1 choice  How many ways to place all 4 digits?  4 x 3 x 2 x 1 = 24 ways

6. Factorial Notation  Many counting and probability calculations involve the product of a series of consecutive integers (i.e. 4x3x2x1)  We can write these products using Factorial Notation  The symbol for this notation is: n! or x!

7. How to Use Factorial Notation  For all natural numbers (integers > 0) n! represents the product of all natural numbers less than or equal to n n! = n x (n-1) x (n-2) … x 3 x 2 x 1  i.e. 5! = 5 x 4 x 3 x 2 x 1 = 120

8. Rules for Factorial Notation  0! = 1  n!/n! = 1  n!/0! = n!/1 = n!