Bell Ringer

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

1. 6
2. 720
3. 24
4. 126
5. 12
6. 22
7. 20
8. 336
9. 720

• 9
• 1320
• 380
• Word Problems
• 5040
• 120
• 6

3
Simple Counting Techniques

• PSD 404 Exhibit knowledge of simple counting
  techniques
• PSD 503 Compute straightforward probabilities
  for common situations

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

1. 2!
2. 3!
3. 4!

2 1 3 2 1 4 3 2 1
2
6
24
This is easy! Give me something harder!
Shut up Xuan! I dont 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!

321 6 ways
54321 120 combinations
7
Working with Factorials

• 4! 3!
• 3! 2!
• 4! 2!

(4321) (321) 30 (321) (21)
4 (4321) (21) 48
6! 4!
(654321) (4321)
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 doesnt
  matter.
• Permutation Arrangement in which order does
  matter.

9
Combinations

• My salad is a combination of lettuce, tomatoes,
  and onions.
• We dont care what order the vegetables are in.
  It could be tomatoes, lettuce, and onions and we
  would have the same salad.
• ORDER DOESNT 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 doesnt 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)!
654321 21(4321)
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)!
7654321 (4321)

210 permutations!
16
Permutation or Combination?

• Eight students were running for student
  government. Two will be picked to represent their
  class.
• Combination It doesnt matter how the two are
  arranged.

8! 2! (8 2)!
87654321 21 (654321)
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)!
87654321 (654321)
8P2

56 ways!