**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.