2.1 Factorial Notation(Textbook Section 4.6)

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)

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

Back to Warm Up Question How many ways can we place the

number 1? 1 can be the 1st, 2nd, 3rd or 4th 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

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

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!

How to Use Factorial Notation For all natural numbers (integers > 0)

n! represents the product of all natural numbers less than or equal to nn! = 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

Rules for Factorial Notation0! = 1n!/n! = 1n!/0! = n!/1 = n!