18.05
Lecture
3
February
7,
2005
n
!
P
n,k
=
(
n
−
k
)!
 choose
k
out
of
n,
order
counts,
without
replacement.
n
k
 choose
k
out
of
n,
order
counts,
with
replacement.
n
!
C
n,k
=
k
!(
n
−
k
)!
 choose
k
out
of
n,
order
doesn’t
count,
without
replacement.
§
1.9
Multinomial
Coeﬃcients
These
values
are
used
to
split
objects
into
groups
of
various
sizes.
s
1
,
s
2
,
...,
s
n

n
elements
such
that
n
1
in
group
1,
n
2
in
group
2,
...,
n
k
in
group
k.
n
1
+
...
+
n
k
=
n
n
±
n
−
n
1
±
n
−
n
1
−
n
2
±
×
...
n
−
n
1
−
...
−
n
k
−
2
±
n
k
±
n
1
n
2
n
3
×
n
k
−
1
n
k
(
n
−
n
1
−
n
2
)!
n
!
(
n
−
n
1
)!
n
3
!(
n
−
n
1
−
n
2
−
n
3
)!
×
...
(
n
−
n
1
−
...
−
n
k
−
2
)!
=
n
1
!(
n
−
n
1
)!
×
n
2
!(
n
−
n
1
−
n
2
)!
×
×
n
k
−
1
!(
n
−
n
1
−
...
−
n
k
−
1
)!
×
1
n
!
n
±
=
=
n
1
!
n
2
!
...n
k
−
1
!
n
k
!
n
1
,
n
2
,
...,
n
k
These
combinations
are
called
multinomial
coeﬃcients.
Further
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 Summer '06
 DrStag
 Probability, times, multinomial coefficients, A1 A2 A3, Ai Aj, disjoint partition

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