chris rents a car for his vacation. he pays $159 for the week and $9.95 for every hour he is late....
TRANSCRIPT
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 9.95h = 5 hours late
Love has no time limits!
1. 62. 7203. 244. 1265. 126. 227. 208. 3369. 720
10. 911. 132012. 380Word Problems10. 504011. 12012. 6
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! =6 • 5 • 4 • 3 • 2 • 1 = 720
This is read as six factorial.
1. 2!2. 3!3. 4!
= 2 • 1 = 3 • 2 • 1= 4 • 3 • 2 • 1
= 2= 6
= 24This 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! =
3•2•1 = 6 ways
5•4•3•2•1 = 120 combinations
4! + 3! =
3! – 2! =
4! 2! =
=
(4•3•2•1) + (3•2•1) = 30
(3•2•1) – (2•1) = 4
(4•3•2•1) (2•1) = 48
6!4!
(6•5•4•3•2•1) (4•3•2•1)
= 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 4-7-2.
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)
nCr =
Where:n = number of things you can choose
fromr = 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
6C2 = = 6!2! (6 – 2)!
6•5•4•3•2•12•1(4•3•2•1)
=30 2 = 15 combinations!
(Order does matter! AB is different from BA)
nPr =
Where:n = number of things you can choose
fromr = number you are choosing
n! (n – r)!
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!
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)!8C2
=
8•7•6•5•4•3•2•12•1 (6•5•4•3•2•1)
=
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)!8P2 =
8•7•6•5•4•3•2•1 (6•5•4•3•2•1)=
= 56 ways!