Show that the conjecture we made in Example is true for five points. Draw a figure that ha...

Show that the conjecture we made in Example is true for five points. Draw a figure that has 31 regions if we are using six points. How was This Example an example of false inductive reasoning?

Example: False Inductive Reasoning

We want to divide a circle into regions by selecting points on its circumference and drawing line segments from each point to each other point. Figure 1.9 shows the greatest number of regions that we get if we have one point (no line segment is possible for this case), two, three, and four points.

Figure Dividing a circle into regions.

Use inductive reasoning to find the greatest number of regions we would get if we had six points on the edge of the circle.

SOLUTION:

This problem seems somewhat similar to what we did in Example, Notice that it appears that each time we add another point, we double the number of regions. It is natural to conjecture that if we have five points, there would be 16 regions, and with six points, we would get 32 regions. However, this is not true. You can try it for yourself by drawing a large circle and picking six points in different ways on the circle. The largest number of regions that you will find is 31, not 32.

Example Determining the Number of Routes for a Salesperson

Suppose that Sharifa is going to visit branch offices of her medical supply company based in Atlanta with branch offices in Boston, New York, Cincinnati, Miami, Detroit, Portland, Los Angeles, Houston, and Kansas City. How many different ways can she begin in Atlanta, visit all branches, and return home?

SOLUTION:

In solving this problem, we will use several of our problem-solving techniques that we discussed in Section 1.1—we will consider simpler examples, draw diagrams, list our examples systematically, and look for a pattern. We will represent the cities by the first letter of their names. Suppose there was only the one office in Atlanta. Then Sharifa would make no visits. If there were one other city—say, Boston—then she would visit Boston and return home. We will represent her one possible trip by a tree diagram as in Figure. The diagram shows that there is only one possible trip: she would travel from A to B. Now suppose that there are two branch offices, Boston and New York. The diagram would now look like Figure. Here we see that there are two possible trips: ABN and ANB, and then return home.

With three branch offices—Boston, New York, and Cincinnati—the diagram would look like Figure.

We begin to see a pattern that we explain in Table. We will indicate a route by listing the first letter of the cities in the order that they are visited.

Figure (a) One branch office, Boston.(b) Two branch offices, Boston and New York.(c) Three branch offices?Boston, New York, and Cincinnati.

Systematically listing routes as more cities are added.