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Bell Ringer

Bell Ringer. Chris rents a car for his vacation. He pays $159 for the week and $9.95 for every hour he is late. When he returned the car, his bill was $208.75. How many hours was he late? 9.95h + 159 = 208.75 - 159 - 159 9.95h = 49.75

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Bell Ringer

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  1. Bell Ringer Chris rents a car for his vacation. He pays $159 for the week and $9.95 for every hour he is late. When he returned the car, his bill was $208.75. How many hours was he late? 9.95h + 159 = 208.75 - 159 - 159 9.95h = 49.75 9.95 9.95 h = 5 hours late Love has no time limits!

  2. Homework • 9 • 1320 • 380 Word Problems • 5040 • 120 • 6 • 6 • 720 • 24 • 126 • 12 • 22 • 20 • 336 • 720

  3. PSD 404: Exhibit knowledge of simple counting techniques* PSD 503: Compute straightforward probabilities for common situations Simple Counting Techniques

  4. Factorials Factorials – The product of the numbers from 1 to n. n! n •(n – 1)•(n – 2)… 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720 This is read as six factorial.

  5. Factorials = 2 • 1 = 3 • 2 • 1 = 4 • 3 • 2 • 1 = 2 • 2! • 3! • 4! = 6 = 24 This is easy! Give me something harder! Shut up Xuan! I don’t want anything harder!

  6. Factorials Factorials are a way to count how many ways to arrange objects. “How many ways could you arrange 3 books on a shelf?” 3! = “How many combinations could you make from 5 numbers?” 5! = 3•2•1 = 6 ways 5•4•3•2•1 = 120 combinations

  7. Working with Factorials (4•3•2•1) + (3•2•1) = 30 (3•2•1) – (2•1) = 4 (4•3•2•1) (2•1) = 48 • 4! + 3! = • 3! – 2! = • 4! 2! = • = 6! 4! (6•5•4•3•2•1) (4•3•2•1) = 30

  8. Permutations & Combinations • Both are used to describe the number of ways you can choose more than one object from a group of objects. The difference in the two is whether order is important. • Combination – Arrangement in which order doesn’t matter. • Permutation – Arrangement in which order does matter.

  9. Combinations • “My salad is a combination of lettuce, tomatoes, and onions.” • We don’t care what order the vegetables are in. It could be tomatoes, lettuce, and onions and we would have the same salad. • ORDER DOESN’T MATTER!

  10. Permutations • “The combination to the safe is 472.” • We do care about the order. “724” would not work, nor would “247”. It has to be exactly 4-7-2. • ORDER DOES MATTER!

  11. Permutations & Combinations • If we had five letters (a, b, c, d, e) and we wanted to choose two of them, we could choose: ab, ac, ad, … • If we were looking for a combination, “ab” would be the same as “ba” because the order would not matter. We would only count those two as one. • If we were looking for a permutation, “ab” and “ba” would be two different arrangements because order does matter.

  12. Combinations (Formula) (Order doesn’t matter! AB is the same as BA) nCr = Where: n = number of things you can choose from r = number you are choosing n! r! (n – r)!

  13. Combinations (Formula) • There are 6 pairs of shoes in the store. Your mother says you can buy any 2 pairs. How many combination of shoes can you choose? So n = 6 and r = 2 6C2 = = 6! 2! (6 – 2)! 6•5•4•3•2•1 2•1(4•3•2•1) 30 2 = = 15 combinations!

  14. Permutations (Formula) (Order does matter! AB is different from BA) nPr = Where: n = number of things you can choose from r = number you are choosing n! (n – r)!

  15. Permutations (Formula) • In a 7 horse race, how many different ways can 1st, 2nd, and 3rd place be awarded? So n = 7 and r = 3 7P3 = = 7! (7 – 3)! 7•6•5•4•3•2•1 (4•3•2•1) = 210 permutations!

  16. Permutation or Combination? Eight students were running for student government. Two will be picked to represent their class. Combination – It doesn’t matter how the two are arranged. 8! 2! (8 – 2)! 8•7•6•5•4•3•2•1 2•1 (6•5•4•3•2•1) 8C2 = = 56 2 = = 28 ways!

  17. Permutation or Combination? Eight students were running for student government. Two will be picked to be president and vice president. Permutation – It matters who is president and who is vice president! 8! (8 – 2)! 8•7•6•5•4•3•2•1 (6•5•4•3•2•1) 8P2 = = = 56 ways!

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