2
Arrange n symbols: r
1
of type 1, r
2
of type 2, …, r
k
of type k
n
r
1
nr
1
r
2
…
n  r
1
 r
2

…
 r
k1
r
k
n!
(nr
1
)!r
1
!
(nr
1
)!
(nr
1
r
2
)!r
2
!
=
…
=
n!
r
1
!r
2
! … r
k
!
How many ways to
rearrange the letters in the
word
“CARNEGIEMELLON”
?
14!
2!3!2!
= 3,632,428,800
Multinomial Coefficients
!
!...r
r
!
r
n!
n
r
...
r
r
if
0,
r
;...;
r
;
r
n
k
2
1
k
2
1
k
2
1
Four ways of choosing
We will choose 2letters word from the
alphabet (L,U,C,K,Y}
1)
no repetitions,
the order is NOT important
LU = UL
2
5
Four ways of choosing
We will choose 2letters word from the
alphabet (L,U,C,K,Y}
2) P(5,2)
no repetitions,
the order is important
LU != UL
P(n,r)=n*(n
1)*…*(n
r+1)
Four ways of choosing
We will choose 2letters word from the
alphabet (L,U,C,K,Y}
3) 5
2
=25 with repetitions,
the order is important