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15.1 Factorial & Fundamental Counting Principles

15.1 Factorial & Fundamental Counting Principles. Factorial. ! factorial notation. 5! 3! 7! 0!. = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1. = 120. = 3 ∙ 2 ∙ 1. = 6. = 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1. = 5040. = 1. Example 1. Faster: count down until reach # in denom.

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15.1 Factorial & Fundamental Counting Principles

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  1. 15.1 Factorial & Fundamental Counting Principles

  2. Factorial ! factorial notation 5! 3! 7! 0! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120 = 3 ∙ 2 ∙ 1 = 6 = 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 5040 = 1

  3. Example 1 Faster: count down until reach # in denom You MUST know how to do this without a calculator!

  4. Example 2 3 2

  5. Example 3 1! = 1 6 7! = 7 5 4 3 2 1 ∙ ∙ ∙ ∙ ∙ ∙ n n–1 n–6 n–3 n–4 n–5 n–2

  6. Example 4 8 6 9 7 5 4 3 2 1 n n–1 n–6 n–3 n–4 n–5 n–2 n+1 n+2

  7. 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

  8. Fundamental Counting Principles What about a day when you don’t care about wearing pants OR shorts? 25 pants 12 shorts How many ways? 37 When doing this AND that – you MULTIPLY When doing this OR that – you ADD

  9. 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

  10. 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? MULT = 55 ADD = 16 you pick 1, how many left to choose from? MULT = 5∙4 = 20

  11. Example 7 Vowels = 5 Consonants = 2 Total = 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 5 + 2 = 7 7 ∙ 6 ∙ 5 ∙ 4 = 840 Have 7 choices for 1st letter , 6 choices for 2nd letter, …

  12. Homework 11-8 Worksheet & Exercise 15-1 WS

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