11.1 Factorial & Fundamental Counting Principles.

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Presentation transcript:

11.1 Factorial & Fundamental Counting Principles

Factorial ! factorial notation 5! 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

Example 2 23

Fundamental Counting Principles You 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 multiply 8∙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 MULTIPLY When 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

11.1 Continued with Permutations

Permutations group of elements with order – ex: zip codes, phone #s key words: – arrange – order – assign order matters!

I have 5 positions and 5 people to fill the positions Make 5 blanks: For 2 nd slot? 5How many choices for 1 st slot? 4 54 ∙ = 5! 321 ∙∙∙ = 120

I have 4 people to choose from to fill 2 positions Make 2 blanks: For 2 nd slot? 4How many choices for 1 st slot? 3 43= 12 ∙

Permutations Formula Formula: # of people/things to choose from # you are arranging Memorize!!

Example

Permutations Formula Last example using formula: (4 people to choose from to fill 2 positions) can put in calc

8 cars parallel parked along one side of street. How many ways can all 8 be arranged? Three diff ways to solve: 2) 8! 3) 6 How many if only 5 cars out of 8 arranged? 87 ∙ = 40, 320 1) 5 4 ∙ 3 ∙ 21 ∙ ∙∙∙