1.
Sample
Space
and
Probability
Part
V:
Coun8ng‐2
ECE
302
Fall
2009
TR
3‐4:15pm
Purdue
University,
School
of
ECE
Prof.
Ilya
Pollak
Par88ons
•  How
many
ways
are
there
to
divide
n
items
into
r
groups
so
that
the
i‐th
group
has
ni
items
(where
n1
+
n2
+
…
+
nr
=
n)?
•  Example:
if
r
=
2,
the
answer
is
€
n
  ways of forming the first group
 n1 
n − n1 objects remain
 n − n1 

 ways of forming the second group
n
 2 
n − n 2 objects remain

 n   n − n1 
 n − n1 −… − n r−1 
Answer :   
 ⋅…⋅ 

nr
 n1   n 2 


n!
(n − n1 )!
(n − n1 −… − n r−1 )!
=
⋅
⋅…⋅
n1!(n − n1 )! n 2!(n − n1 − n 2 )!
n r!(n
− n
−
n r−1
1 −…
r )!


−n
= 0!=1
=
n!
n1!n 2!…n r!


n
These are denoted 
 and called multinomial coefficients.
 n1,n 2 ,…,n r 
The reason for this name is that

 n n
n
n
( x1 + x 2 + … + x r ) =
∑  n ,n ,…,n x1 1 x 2 2 ⋅…⋅ x rnr
1
2
r
n1 +n 2 +…+n r = n
Example
(p.
68)
•  90
students,
including
Joe
and
Jane,
are
to
be
split
into
3
classes
of
equal
size,
at
random.
•  What
is
the
probability
that
Joe
and
Jane
end
up
in
the
same
class?
Example
•  90
students,
including
Joe
and
Jane,
are
to
be
split
into
3
classes
of
equal
size,
at
random.
•  What
is
the
probability
that
Joe
and
Jane
end
up
in
the
same
class?
Example
•  90
students,
including
Joe
and
Jane,
are
to
be
split
into
3
classes
of
equal
size,
at
random.
•  What
is
the
probability
that
Joe
and
Jane
end
up
in
the
same
class?
Example
•  90
students,
including
Joe
and
Jane,
are
to
be
split
into
3
classes
of
equal
size,
at
random.
•  What
is
the
probability
that
Joe
and
Jane
end
up
in
the
same
class?
A
simpler
method
•  Place
Joe
in
one
class.
A
simpler
method
•  Place
Joe
in
one
class.
•  For
Jane,
there
are
89
total
possibili8es,
of
which
29
put
her
in
the
same
class
with
Joe.
A
simpler
method
•  Place
Joe
in
one
class.
•  For
Jane,
there
are
89
total
possibili8es,
of
which
29
put
her
in
the
same
class
with
Joe.
•  Answer:
29/89.