M
TH
355 N
OTES
:
Oct 18, 2001
Some Notes on Counting
cells
objects
specification
number
notes
n
, dist
m
, dist
Place all objects in the
cells. Some cells may be
empty.
n
m
For each object choose a cell in
n
ways to put it in.
n
, dist
m
types
Place exactly one object
in each cell. Assume we
have an unlimited num
ber of each type.
m
n
For each cell we have
m
choices
of type.
n
, dist
m
, dist
m
k
objects in
k
th
cell,
k
= 1
· · ·
, n
. Here
m
=
m
1
+
· · ·
+
m
n
.
m
!
m
1
!
· · ·
m
n
!
Place the objects in order, then
permute the contents of each
cell.
n
, indist
m
,
dist,
m
=
m
1
+
· · ·
+
m
n
m
k
objects in
k
th
cell.
Let
p
i
,
i
=
1
, . . . , h
be
the
number
of
oc
curences of each distinct
value of the
m
k
.
m
!
m
1
!
· · ·
m
n
!
1
p
1
!
· · ·
p
h
!
Permute
the
cells
having
the
same cardinality since we can’t
distinguish them.
n
,
dist,
n
≥
m
m
,
m
=
m
1
+
· · ·
+
m
h
m
k
objects of type
k
.
Place all objects, at most
one per cell.
m
!
m
1
!
· · ·
m
h
!
n
m
Choose successively the cells to
receive an object of type
k
, in
cluding
n
−
m
objects of “vir
tual” type.
n
, dist
n
, dist
Place one object in each
cell.
n
!
Think of the cells as being in or
der.
n
, dist
m
,
dist,
m
≤
n
Place at most one object
in each cell.
n
!
(
n
−
m
)!
Introduce
n
−
m
virtual distin
guishable object in
(
n
−
m
)!
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 Spring '08
 McAllister
 Math, Counting, cells, Natural number

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