In this lesson, our instructor Vincent Selhorst-Jones teaches all about Counting. Youll learn what combinatorics means and the definition of an event. Vincent will show you how to visualize an event with a branching line diagram, as well as, the addition, multiplication, and pigeonhole principles. Youll draw pictures and solve problems alongside Vincent as he works through five examples on screen.
An event is something happening that can occur some number of different ways. If we want, we can denote an event with a symbol (such as E) and similarly for the number of ways (such as n). [Some teachers and textbooks will not use the word `event'
and might use another word or nothing at all to describe this idea. That's fine. The important part is just to be aware of certain things having multiple possible outcomes or choices. All the coming ideas will work fine in any case; `event' is just a
way to describe this formally.]
We can visualize an event (or series of events, as we'll soon see) with a branching line diagram. At each event in the diagram, it splits into its various possible outcomes.
Addition Principle: Let there be two events E1 and E2, where E1 can occur in m ways and E2 can occur in n. If precisely one of the events will occur (not both), the total possible ways for something
to occur is
Multiplication Principle [AKA Fundamental Counting Principle]: Let there be two events E1 and E2, where E1 can occur in m ways and E2 can occur in n. If both events will occur (and they have no effect
on each other), the total possible ways for something to occur is
Pigeonhole Principle: Let there be some event E that can occur in n ways. If the event E happens n+1 times, then at least one of the ways E can occur must happen twice (or more). Equivalently, given n boxes and n+1 objects, if all the objects are
put into the boxes, then at least one box must have (at least) two objects.
When working on counting problems, it can help immensely to draw pictures and diagrams. Being able to visually follow the events and choices will make it much easier to solve problems.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.