, an r-permutation of n objects is a permutation
of r of them, that is, an arrangement of r of the objects in some
linear order.Author: Warren Odgers SID: 7631
Permutations
Definition: A permutation of a set of distinct objects is an
arrangement of them in some linear order. Since the objects are distinct, the
order is important. For natural numbers n and r, with , an r-permutation of n objects is a permutation
of r of them, that is, an arrangement of r of the objects in some
linear order.
Mathematically speaking, for integers n,r
, 
, 
, the function 
, whose base case is defined as
is:
The number of permutations of n objects is n!. The number
of
r-permutations of n objects is
. Since the base case is
, the unique 0-permutation is called the empty
permutation.
Example:
Suppose that in a certain computer science class, there were 20 students each of whom were required to construct a small robot capable of locating and carrying a ball. Given that the professor can only grade one at a time, all of the students complete the project, and none of the students are given priority grading, how many ways may the robot projects be presented?
Answer:
Since each of the students has a project, there are n=20 places in the
presentation order. Also, there are r=20 distinct robots to present.
Therefore there are 
ways the projects may be presented. How many is that?
20!=2,462,902,008,176,640,000
There are 2.4 million billion (2.4 quintillion) ways to order the students’ projects.
Exercise:
Suppose that there are 500 tickets, numbered 1, 2, 3, …, 500, that are sold to 500 different people for a prize drawing. Six different prizes are awarded, including a grand prize ($1,000,000). How many ways are there to award the prizes if the people holding tickets 192 and 399 both win prizes?
For more on combinatorics, visit The Combinatorics Net or The Mathematics Archives
