Transcript 2.1 Factorial Notation - Halton Catholic District School Board

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?


IF we have placed numbers 1, 2, and 3, how
many ways can we place the number 4?


2 remaining digit places, so 2 ways
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 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
Rules for Factorial Notation
0!
=1
n!/n! = 1
n!/0! = n!/1 = n!