I'm trying to understand the addition and multiplication principles in counting.

So far I understand that the addition principle is used to count the number of ways in which one of any events can occur while the multiplication principle is to count the number of way in which many events will occur together.

However, many definitions make a point of using the words "independent events" or "mutually exclusive events". What is meant by these words?

Consider a coloring problem, what are the ways to color a map so adjacent states are not the same color. This is a multiplication problem because we are calculating all the ways in which it can be colored. However, the coloring of a state is not independent of the coloring of another state, as the color of each state depends on the colors of adjacent states.

  • $\begingroup$ Counting the ways a map can be colored so that adjacent states are not the same color usually cannot be done with just the Multiplication Principle. The restrictions on the coloring of adjacent states can be addressed by the Inclusion-Exclusion Principle. $\endgroup$ – N. F. Taussig Nov 3 '18 at 9:34

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