LEARNING GOALS

1. LEARNING GOALS • I understand the contents of the Highland Secondary School Mathematics Department Policies • I understand the use of graphs to help solve counting problems • I can identify areas of improvement on the review of Prerequisite Skills

2. 4.1 Organized Counting - HISTORY • A frequently asked question there at the time was "Is it possible to take a walk through town, starting and ending at the same place, and cross each bridge exactly once?"

3. 4.1 Organized Counting combinatorics - sometimes called the science of counting, the branch of mathematics concerned with the selection, arrangement, and combinations of objects within a finite set of objects. Combinatorial theory deals with: • existence (does a particular arrangement exist?) • enumeration (how many such arrangements are there?) • structure (what are the properties of each arrangement?)

4. 4.1 Organized Counting - HISTORY • Leonhard Euler (1701-1783) was a Swiss mathematician who spent most of his life in Russia. He was responsible for making a number of the initial contributions to combinatorics both in graph theory and enumeration. • One of these contributions was a paper he published in 1736. The people of an old town in Prussia called Königsberg (now Kaliningrad in Russia) brought to Euler's attention a stirring question about moving along bridges. • Euler wrote a paper answering the question called "The Seven Bridges of Königsberg." The town was on an island in the Pregel river and had seven bridges.

5. 4.1 Organized Counting - HISTORY • Euler generalized the problem to points and lines where the island was represented by one point and the bridges were represented by lines. By abstracting the problem, Euler was able to answer the question. It was impossible to return to the same place by only crossing each bridge exactly once. • The abstract picture he drew of lines and points was a graph, and the beginnings of graph theory.

6. 4.1 Organized Counting - HISTORY

8. 4.1 Organized Counting • It has applications in such diverse areas as managing computer and telecommunication networks, predicting poker hands, dividing tasks among workers, and finding a pair of socks in a drawer. • Combinatorics has its roots in the 17th- and 18th-century attempts to analyze the odds of winning at games of chance. • The advent of computers in the 20th cent. made possible the high-speed calculations required to analyze the multitude of possibilities involved in combinatory methods to solving problems. • Branches of combinatorics include graph theory and combinations and permutations.

9. 4.1 Organized Counting Making a Series of Choices – Process of Steps Example 1 – Travel Itineraries You are planning a trip to Montreal. You would like to stop in Ottawa along the way. Checking the internet, you find that you can take a Bus, Train, or two different flights From Hamilton to Ottawa, then, you have a choice of Train or Bus from Ottawa to Montreal. How many ways can you travel from Hamilton to Montreal? Example 2 – Choosing Letters or Numbers In Alberta, a license plate consists of 3 letters first and then 3 numbers. How many different license plates are available in Alberta?

10. 4.1 Organized Counting Tree Diagram for Example 1- Travel Itineraries Train Bus Bus Train Train 8 Possible ways To travel from Hamilton to Montreal Bus Train Flight #1 Bus Train 4 Choices x 2 Choices Flight #2 Bus HAMILTON to OTTAWA OTTAWA to MONTREAL STEP 2 STEP 1 PROCESS

11. 4.1 Organized Counting Solution for Example 2 – Alberta License Plates 26 26 x 26 x 10 x 10 x 10 x PROCESS LETTER LETTER LETTER NUMBER NUMBER NUMBER = 17576000 These types of counting problems illustrate the: Fundamental (Multiplicative) Counting Principle If a task or process is made up of stages with separate choices, the total number of choices is m x n x p x …, where m is the number of choices for the first stage, n is the number of choices for the second stage, p is the number of choices for the third stage, and so on.

12. 4.1 Organized Counting Applying the Fundamental Counting Principle – INDIRECT METHOD Jeff is a cross country athlete that has five pairs of running shoes in his gym bag. In how many ways can he pull out two unmatched shoes one after another without looking? You can find the number of ways of picking unmatched shoes by subtracting the number of ways of picking matching ones from the total number of ways of picking ANY two shoes. Total ways of picking ANY two shoes: 10 9 = 90 X FIRST SHOE SECOND SHOE 5 x 2 = 10 Ways of picking MATCHING shoes: 90 – 10 = 80 Therefore, number of ways of picking unmatched shoes is:

13. 4.1 Organized Counting Counting Subsets of Possibilities Separately – Mutually Exclusive Events Sailing ships used to send messages with signal flags flown from their masts. How many different signals are possible with a set of 4 distinct flags if a minimum of two flags is used for each signal? Case #1 - Signals with two flags: 4 x 3 = 12 Case #2 - Signals with three flags: 4 x 3 x 2 = 24 Case #3 - Signals with four flags: 4 x 3 x 2 x 1 = 24 TOTAL: (add them) = 60 We have three mutually exclusive events, (you cannot send a message with three flags AND four flags at the same time) thus we add the cases together to get the total. This is called the Additive Counting Principle or the Rule of Sum. If one mutually exclusive action can occur in m ways, and a second in n ways, and a third in p ways, and so on, then there are m + n + p + … ways in which only one of these actions can occur.

14. 4.1 Organized Counting WORK ON: Page 229 #1,2,3,4,5a,6,8,9,11,12,13,16,18,19,22,23. The more you practice, the better you will get!