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Chapter 1 summary
Combinatorial Analysis
•
Fundamental counting principles:
1.
(multiplication principle)
Say task 1 can be done in
n
1
ways, task 2 can be done in
n
2
ways,
..., task
k
can be done in
n
k
ways. Then if we wish to do all the tasks in sequence, i.e. task 1,
followed by task 2, .
.. , followed by task
k
(choosing just one of the available ways to do each of
the tasks),
then this can be done in n
1
·
n
2
·
. . .
·
n
k
ways
.
2.
(addition principle)
Say we have just one task to do. If we have
m
ways to do it and also
n
ways to do it (in which there is no overlap between the two sets of ways of doing it),
then the
one task can be done in m
+
n ways
.
•
Permutations and combinations:
Say we have
n
diﬀerent objects and we wish to select
k
of those
objects.
1.
(permutations)
If we then wish to arrange those
k
objects in a row (so each diﬀerent ordering
of the objects in a row counts as a diﬀerent selection) then there are
P
(
n, k
) =
n
(
n

1)(
n

2)
. . .
(
n

k
+ 1)
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