The Generalized Pigeonhole Principle

Pigeonhole Principle

Permutations

Let us give you the following problem : "How many ways are there to arrange 5 boys and 3 girls in a row so that no two
girls are adjacent?" This is a typical example of a more general problem of arranging some distinct objects subject to certain additional conditions.

 Two Basic Counting Principles

      In our everyday lives, we often need to enumerate "events" such as, the arrangement of objects in a certain way, the partition of things under a certain condition, the distribution of items according to a certain  specification, and so on. For instance, we may come across counting problems of the following types:

      "How many ways are there to arrange 5 boys and 3 girls in a row so that no two girls are adjacent?"

      "How many ways are there to divide a group of 10 people into three groups consisting of 4, 3 and 2 people respectively with 1 person rejected?"