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.

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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 $ How many hours was he late? 9.95h = h = h = 5 hours late Love has no time limits!

Word Problems

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)… 6! = = 720 This is read as six factorial.

1. 2! 2. 3! 3. 4! = 2 1 = = = 2 = 6 = 24 This is easy! Give me something harder! 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! = “How many combinations could you make from 5 numbers?” 5! = 321 = 6 ways = 120 combinations

 4! + 3! =  3! – 2! =  4! 2! =  = (4321) + (321) = 30 (321) – (21) = 4 (4321) (21) = 48 6! 4! (654321) (4321) = 30

 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.

 “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!

 “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  ORDER DOES MATTER!

 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.

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

 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 2 = = 6! 2! (6 – 2)! (4321) = 30 2 =15 combinations!

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

 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 3 = = 7! (7 – 3)! (4321) =210 permutations!

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 C 2 = (654321) = 56 2 = = 28 ways!

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 P 2 = (654321) = = 56 ways!