Bell Ringer Chris rents a car for his

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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. 95 h + 159 = 208. 75 - 159 9. 95 h = 49. 75 9. 95 h = 5 hours late Love has no time limits!

Homework 1. 2. 3. 4. 5. 6. 7. 8. 9. 6 720 24 126 12 22 20 336 720 9 1320 380 Word Problems 10. 5040 11. 120 12. 6 10. 11. 12.

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

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

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

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

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

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.

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!

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!

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.

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

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 6 C 6! 6 • 5 • 4 • 3 • 2 • 1 2 = 2! (6 – 2)! = 2 • 1(4 • 3 • 2 • 1) 30 = = 15 combinations! 2

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

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

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! 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 8 C 2 = 2! (8 – 2)! = 2 • 1 (6 • 5 • 4 • 3 • 2 • 1) 56 = = 28 ways! 2

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 • 7 • 6 • 5 • 4 • 3 • 2 • 1 8 P 2 = (8 – 2)! = (6 • 5 • 4 • 3 • 2 • 1) = 56 ways!

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