15.1 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 You MUST know how to do this without a calculator!

Example 2 23

n n–1 n–2 n–3n–4n–5 n– Example 3 7! = ∙ ∙∙∙∙∙ 1! = 1

n n–1 n–2 n–3n–4n–5 n– Example 4 n+1 n+2 8 9

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

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 = 5 Consonants = 2 Total = = 7 Have 7 choices for 1 st letter 7, 6 choices for 2 nd letter, … 654 ∙∙∙ = 840

Homework 11-8 Worksheet & Exercise 15-1 WS