CSCE 235
Handout 20: Combinatorics
Assigned
March 10, 2004
PERMUTATIONS
ORDER IS IMPORTANT
Definition:
A permutation of a set of
distinct symbols is an arrangement of them in a line in some order. For natural numbers n and r, with
, an r-permutation of n symbols is permutation of r of
them, that is, an arrangement of r of the symbols in a line in some
order.
Definition: For integers n and r, 
, 
, the symbol 
is defined by 
and
Given natural numbers n
and r, with 
, the number of ways to place r marbles of different
colors into n numbered boxes, at most one marble to a box, is 
The number of
permutations of n symbols is n!. The number of r-permutations of n symbols is 
. It is also
convenient to decree 
, the unique 0-permutation being the empty permutation.
COMBINATIONS
ORDER IS NOT IMPORTANT ONE PER BOX
Definition: A combination of a set of objects is a subset of them. A subset of r objects is called an r-combination or a combination of the objects taken r at a time. Or, we also say that an r-combination of elements of a set is an unordered selection of r elements from the set.
Definition:
For integers n and r, 
, 
, the number of ways to choose r objects from n
objects is 
. The number of r-combinations of n objects is 
(read “n
choose r”) or sometimes also denoted as 
. The binomial coefficient 
is defined by 
.
REPETITIONS
ORDER IS NOT IMPORTANT ANY PER BOX
The number of
ways to put r identical marbles into n boxes is 
.
DERANGEMENTS
Definition: A derangement of n distinct symbols
which have some natural order is a permutation in which no symbol is in its
correct position. The number of
derangement of n symbols is denoted 
. The number
of derangements of 
ordered symbols is
.
THE
BINOMIAL THEOREM
For any a and b and any natural number n,
.
This can be proved by induction using the following recursion relation:
for 
.
RULES
OF THUMB
|
Symbols |
Uses |
|
n! |
* Number of permutations of n distinct symbols |
|
|
* Number of ways to put r different colored marbles into n numbered boxes (order is important), at most one to a box * Number of permutations of n distinct symbols used r at a time |
|
|
* Number of ways to put r identical marbles into n boxes (order is not important), at most one to a box * Number of combinations of n distinct symbols used r at a time * Number of ways to
choose r objects from n |
|
|
* Number of ways to put r identical marbles into n boxes (order is not important), any number in a box * Combinations with repetition: the number of r-combinations from a set with n elements when repetition of elements is allowed |
|
|
* Number of ways to put r distinct marbles into n boxes (order is not important), any number in a box * Permutations with repetition: the number of r-permutations of a set of n objects with repetition allowed |
Based on (Rosen 2003), (Goodaire and Parmenter 2002), and (Ross and Wright 1988).