1. 6.2 Fundamental Counting Principal

2. Introduction
Joe Futz has two pairs of pants (one plaid, one striped) and three shirts (one red, one yellow, and one green). An outfit is one pair of pants and one shirt. If Joe has absolutely no sense of what looks well together, how many outfits does he have to choose from?

3. Tree Diagrams Fundamental Counting Principal:
two pairs of pants (one plaid, one striped) and three shirts (one red, one yellow, and one green).
Results:
Plaid, Red
Plaid, Yellow
Plaid, Green
Striped, Red
Striped, Yellow
Striped, Green R
Fundamental Counting Principal:
If a first task has m possible completions, a second has n possible completions, and a final task has q possible completions, then the number of possible solutions for all the tasks is 𝑚∙𝑛∙𝑞 P Y G GO R S Y G

4. Vocab
On the following tree diagram, Go, P, Y, etc. mark nodes of the diagram.
“Go” is the starting node.
R, Y, G ending (terminal) nodes.
The lines from Go to P, P to R, S to Y, etc. (linking consecutive nodes) are branches.
A sequence of branches go from Go to an ending node is a path. Each path shows a possible option.

5. Example:
Suppose Joe chooses a shirt first, then pants. Construct a tree diagram and list the set of possible outcomes.
“two pairs of pants (one plaid, one striped) and three shirts (one red, one yellow, and one green)”

6. Practice Problem
Joe’s wealthy cousin Plato has 14 pairs of pants, 37 shirts and an incredible collections of 118 hats, no two alike. Just think about the tree diagram, and determine the number of Plato’s possible outfits if an outfit consists of the following items, chosen in the order named:
Pants and shirt
Shirt, pants, and hat
Hat and pants
Shirt and pants
14∙37
14∙37∙118
118∙14
37∙14

7. Practice Problem
You must design the “standard office electronic workstation” for your company. Each station consists of a microcomputer, a keyboard, a video monitor, a printer, and a modem. You are considering two computers, three keyboards, three monitors, four printers, and two modems. How many workstation designs are possible?
144

8. Practice Problem
A cafeteria offers three salads, four main dishes, two desserts, and five beverages.
If you must make one choice from each category, how many distinct (non-identical) meal choices are possible?
If you many make at most one choice from each category (and choosing “nothing” is a possibility in each case), how many non-identical meal choices are possible?
3∙4∙2∙5=120
4∙5∙3∙6=360

9. Practice Problem
A license plate consists of three alphabet letters followed by three digits (no spaces). How many plates are possible?
What if repeated letters or numbers are not permitted?
26∙26∙26∙10∙10∙10=17,576,000
26∙25∙24∙10∙9∙8=11,232,000

10. Example
Form a 3-digit numeral using digits from {2, 4, 5, 7, 9}, with no digit being repeated.
How many such numerals are possible?
How many are numbers less than 500?
How many are even numbers?
How many are multiples of 5?

11. Homework
Page 14 #19-34