Combinations 組合

In general, as represents the number of different ways that a group of r items could be selected from n items when the order of selection is relevant, and, as each group of r items will be counted r! times in this count, it follows that the number of different groups of r items that could be formed from a set of n items is

Notation and Terminology
We define , for , by and say that represents the number of possible combinations of n objects taken r at a time.

Example
A committee of 3 is to be formed from a group of 20 people. How many different committees are possible?
Solution:
There are possible committees.

A useful combinatorial identity is

Consider a group of n objects and fix attention on some particular one of these objects -- call it object 1. Now, there are combinations of size r that contain object 1 (since each such combination is formed by selecting r-1 from the remaining n-1 objects). Also, there are combinations of size r that do not contain object 1. As there is a total of combinations of size r.