16
Counting (cont.)
16.1
Solving problems by applying basic principles
•
often there are restrictions
e.g.
third letter must be A or E
first digit must not be zero
usually best to deal with restrictions first
often helpful to use boxes
•
e.g.
arrangements of all letters of BLISTER
no restriction
start with B
end with a vowel
•
e.g.
3digit numbers from digits 02468
implied restriction: first cannot be zero
•
another common restriction
values must be adjacent
e.g.
arrangements of 12345 in which 1 and 2 are adjacent
•
another common variation
elements in a circle
illustrate with ABC in a circle
only 2 arrangements
generally
use one value as a marker
arrange others relative to the marker
16.2
The addition or sum rule
•
If one operation can be performed in
m
ways or a second operation can be performed in
n
ways, and
only one of the operations can be performed
, then the number of possible operations
is
m
+
n
•
in set terms
given sets
A
and
B
with
A
∩
B
=
∅
,

A
∪
B

=

A

+

B

•
this rule is often useful with a complement of a restriction
•
e.g.
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 Fall '06
 Carter
 Combinatorics, Permutations, adjacent e.g. arrangements

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