Background color for the day is fuchsia.
This section of the text considers 3 specific counting problem types. While not particularly important in themselves, the thinking patterns these problems force one to adopt have wide applicability.
Toss a coin N times. If, e.g., N=4, there are 4 possible outcomes which could be listed as HH, HT, TH, and TT. In general, there are 2 raised to the Nth power possible outcomes. This can be seen using the multiplication principle.
Example: Toss a coin 7 times.
Many problems can be viewed as coin tossing problems in disguise.
Thinking of the lines in the picture as representing city streets, how many different routes are there from A to B? (Backtracking is not allowed.) Alogether one must take 7 eastward steps and 4 southward steps, so the only thing that can vary is the order in which these are done. The red path could be described as SEESEEESSEE and the yellow path as ESSSEEESEEE.
An urn is a container in which there are some given number of white balls and some given number of black balls. (Actual colors vary from one problem to the next.) Often, the actual balls are indistinguishable except for color, but it almost always useful to imagine that the balls have been uniquely numbered. Urn problems pose various questions about a sample of balls drawn from the urn.
For example, consider problem 5.6.4: An urn contains 12 numbered balls, of which 8 are red and 4 are white. A sample of 4 balls is to be selected.
Almost all counting problems, especially the ones that arise in probability, can be viewed as some form of sampling. Suppose we have n objects (e.g, people) and we select a sample of size r. How many such samples are possible? The answer depends on how the sampling is done: with or without replacement, and with ordering or without ordering.
Once an item is selected, it cannot be selected a second time.
Selected items are returned to the pool and are available to be selected again.
