15.1 factorial & fundamental counting principles
TRANSCRIPT
15.1 Factorial & Fundamental Counting Principles
Factorial
! factorial notation5!
3!
7!
0!
= 3 ∙ 2 ∙ 1
= 120= 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
= 6
= 1
= 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1= 5040
Example 1
Faster: count down until reach # in denom
7!
5!
7 6 5 4 3 2 1
5 4 3 2 1
42
7!
5!
7 6 5!
5!
42
You MUST know how to do this without a calculator!
Example 2
10!
5!5!
10 9 8 7 6 5!
5 4 3 2 1 5!
3 2 7 6
23
6 7 6 252
n n–1 n–2 n–3 n–4 n–5 n–6
7 6 5 4 3 2 1
Example 3
7! = ∙ ∙ ∙ ∙ ∙ ∙
( 1)!
1!( 2)!
n
n
( 1)( 2)!
1!( 2)!
n n
n
1! = 1
1n
n n–1 n–2 n–3 n–4 n–5 n–6
7 6 5 4 3 2 1
Example 4
( 1)!
( 1)!
n
n
( 1)( )( 1)!
( 1)!
n n n
n
( 1)n n
n+1n+2
89
2n n
Fundamental Counting PrinciplesYou have 8 pants & 4 shirts. How many ways can you select a pants-AND-shirt combination?
How many choices?
What did you do to get that?
32
multiply8∙4 = 32
Fundamental Counting Principles
25 pants 12 shorts
What about a day when you don’t care about wearing pants OR shorts?
How many ways? 37
When doing this AND that – you MULTIPLYWhen doing this OR that – you ADD
Example 5
There are 25 dogs and 10 cats.
How many ways to choose:
- a dog or a cat?
- a dog and then a cat?
ADD = 35
MULT = 250
Example 6
There are 11 novels and 5 mysteries.
How many ways to choose:
- a novel and then a mystery?
- a novel or a mystery?
- a mystery and then another mystery?
ADD = 16
MULT = 55
you pick 1, how many left to choose from?
MULT = 5∙4 = 20
Example 7
Using the letters in SEQUOIA.How many ways to choose:• a vowel and a consonant?• a vowel or a consonant?• 4-letter “words” using no letter more than
once in a “word”?
5∙2 = 10
Vowels = 5Consonants = 2Total = 7
5 + 2 = 7
Have 7 choices for 1st letter
7
, 6 choices for 2nd letter, …
6 5 4∙ ∙ ∙ = 840
Homework
11-8 Worksheet &
Exercise 15-1 WS