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Theorem
edit
For any positive integer m and any non-negative integer n , the multinomial formula describes how a sum with m terms expands when raised to an arbitrary power n :
(
x
1
+
x
2
+
⋯
+
x
m
)
n
=
∑
k
1
+
k
2
+
⋯
+
k
m
=
n
k
1
,
k
2
,
⋯
,
k
m
≥
0
(
n
k
1
,
k
2
,
…
,
k
m
)
∏
t
=
1
m
x
t
k
t
,
{\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{\begin{array}{c}k_{1}+k_{2}+\cdots +k_{m}=n\\k_{1},k_{2},\cdots ,k_{m}\geq 0\end{array}}{n \choose k_{1},k_{2},\ldots ,k_{m}}\prod _{t=1}^{m}x_{t}^{k_{t}}\,,}
where
(
n
k
1
,
k
2
,
…
,
k
m
)
=
n
!
k
1
!
k
2
!
⋯
k
m
!
{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}
is a multinomial coefficient . The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n . That is, for each term in the expansion, the exponents of the x i must add up to n . Also, as with the binomial theorem , quantities of the form x 0 that appear are taken to equal 1 (even when x equals zero ).
In the case m = 2 , this statement reduces to that of the binomial theorem.
Example
edit
The third power of the trinomial a + b + c is given by
(
a
+
b
+
c
)
3
=
a
3
+
b
3
+
c
3
+
3
a
2
b
+
3
a
2
c
+
3
b
2
a
+
3
b
2
c
+
3
c
2
a
+
3
c
2
b
+
6
a
b
c
.
{\displaystyle (a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3b^{2}a+3b^{2}c+3c^{2}a+3c^{2}b+6abc.}
This can be computed by hand using the distributive property of multiplication over addition, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
a
2
b
0
c
1
{\displaystyle a^{2}b^{0}c^{1}}
has the coefficient
(
3
2
,
0
,
1
)
=
3
!
2
!
⋅
0
!
⋅
1
!
=
6
2
⋅
1
⋅
1
=
3.
{\displaystyle {3 \choose 2,0,1}={\frac {3!}{2!\cdot 0!\cdot 1!}}={\frac {6}{2\cdot 1\cdot 1}}=3.}
a
1
b
1
c
1
{\displaystyle a^{1}b^{1}c^{1}}
has the coefficient
(
3
1
,
1
,
1
)
=
3
!
1
!
⋅
1
!
⋅
1
!
=
6
1
⋅
1
⋅
1
=
6.
{\displaystyle {3 \choose 1,1,1}={\frac {3!}{1!\cdot 1!\cdot 1!}}={\frac {6}{1\cdot 1\cdot 1}}=6.}
Alternate expression
edit
The statement of the theorem can be written concisely using multiindices :
(
x
1
+
⋯
+
x
m
)
n
=
∑
|
α
|
=
n
(
n
α
)
x
α
{\displaystyle (x_{1}+\cdots +x_{m})^{n}=\sum _{|\alpha |=n}{n \choose \alpha }x^{\alpha }}
where
α
=
(
α
1
,
α
2
,
…
,
α
m
)
{\displaystyle \alpha =(\alpha _{1},\alpha _{2},\dots ,\alpha _{m})}
and
x
α
=
x
1
α
1
x
2
α
2
⋯
x
m
α
m
{\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{m}^{\alpha _{m}}}
Proof
edit
This proof of the multinomial theorem uses the binomial theorem and induction on m .
First, for m = 1 , both sides equal x 1 n since there is only one term k 1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m . Then
(
x
1
+
x
2
+
⋯
+
x
m
+
x
m
+
1
)
n
=
(
x
1
+
x
2
+
⋯
+
(
x
m
+
x
m
+
1
)
)
n
=
∑
k
1
+
k
2
+
⋯
+
k
m
−
1
+
K
=
n
(
n
k
1
,
k
2
,
…
,
k
m
−
1
,
K
)
x
1
k
1
x
2
k
2
⋯
x
m
−
1
k
m
−
1
(
x
m
+
x
m
+
1
)
K
{\displaystyle {\begin{aligned}&(x_{1}+x_{2}+\cdots +x_{m}+x_{m+1})^{n}=(x_{1}+x_{2}+\cdots +(x_{m}+x_{m+1}))^{n}\\[6pt]={}&\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+K=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},K}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}(x_{m}+x_{m+1})^{K}\end{aligned}}}
by the induction hypothesis. Applying the binomial theorem to the last factor,
=
∑
k
1
+
k
2
+
⋯
+
k
m
−
1
+
K
=
n
(
n
k
1
,
k
2
,
…
,
k
m
−
1
,
K
)
x
1
k
1
x
2
k
2
⋯
x
m
−
1
k
m
−
1
∑
k
m
+
k
m
+
1
=
K
(
K
k
m
,
k
m
+
1
)
x
m
k
m
x
m
+
1
k
m
+
1
{\displaystyle =\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+K=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},K}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}\sum _{k_{m}+k_{m+1}=K}{K \choose k_{m},k_{m+1}}x_{m}^{k_{m}}x_{m+1}^{k_{m+1}}}
=
∑
k
1
+
k
2
+
⋯
+
k
m
−
1
+
k
m
+
k
m
+
1
=
n
(
n
k
1
,
k
2
,
…
,
k
m
−
1
,
k
m
,
k
m
+
1
)
x
1
k
1
x
2
k
2
⋯
x
m
−
1
k
m
−
1
x
m
k
m
x
m
+
1
k
m
+
1
{\displaystyle =\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+k_{m}+k_{m+1}=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},k_{m},k_{m+1}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}x_{m}^{k_{m}}x_{m+1}^{k_{m+1}}}
which completes the induction. The last step follows because
(
n
k
1
,
k
2
,
…
,
k
m
−
1
,
K
)
(
K
k
m
,
k
m
+
1
)
=
(
n
k
1
,
k
2
,
…
,
k
m
−
1
,
k
m
,
k
m
+
1
)
,
{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m-1},K}{K \choose k_{m},k_{m+1}}={n \choose k_{1},k_{2},\ldots ,k_{m-1},k_{m},k_{m+1}},}
as can easily be seen by writing the three coefficients using factorials as follows:
n
!
k
1
!
k
2
!
⋯
k
m
−
1
!
K
!
K
!
k
m
!
k
m
+
1
!
=
n
!
k
1
!
k
2
!
⋯
k
m
+
1
!
.
{\displaystyle {\frac {n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}}{\frac {K!}{k_{m}!k_{m+1}!}}={\frac {n!}{k_{1}!k_{2}!\cdots k_{m+1}!}}.}
Multinomial coefficients
edit
The numbers
(
n
k
1
,
k
2
,
…
,
k
m
)
{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}}
appearing in the theorem are the multinomial coefficients . They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials :
(
n
k
1
,
k
2
,
…
,
k
m
)
=
n
!
k
1
!
k
2
!
⋯
k
m
!
=
(
k
1
k
1
)
(
k
1
+
k
2
k
2
)
⋯
(
k
1
+
k
2
+
⋯
+
k
m
k
m
)
{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}={k_{1} \choose k_{1}}{k_{1}+k_{2} \choose k_{2}}\cdots {k_{1}+k_{2}+\cdots +k_{m} \choose k_{m}}}
Sum of all multinomial coefficients
edit
The substitution of x i = 1 for all i into the multinomial theorem
∑
k
1
+
k
2
+
⋯
+
k
m
=
n
(
n
k
1
,
k
2
,
…
,
k
m
)
x
1
k
1
x
2
k
2
⋯
x
m
k
m
=
(
x
1
+
x
2
+
⋯
+
x
m
)
n
{\displaystyle \sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}=(x_{1}+x_{2}+\cdots +x_{m})^{n}}
gives immediately that
∑
k
1
+
k
2
+
⋯
+
k
m
=
n
(
n
k
1
,
k
2
,
…
,
k
m
)
=
m
n
.
{\displaystyle \sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}=m^{n}.}
Number of multinomial coefficients
edit
The number of terms in a multinomial sum, #n ,m , is equal to the number of monomials of degree n on the variables x 1 , …, x m :
#
n
,
m
=
(
n
+
m
−
1
m
−
1
)
.
{\displaystyle \#_{n,m}={n+m-1 \choose m-1}.}
The count can be performed easily using the method of stars and bars .
Valuation of multinomial coefficients
edit
The largest power of a prime p that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem .
Asymptotics
edit
By Stirling's approximation , or equivalently the log-gamma function 's asymptotic expansion,
log
(
k
n
n
,
n
,
⋯
,
n
)
=
k
n
log
(
k
)
+
1
2
(
log
(
k
)
−
(
k
−
1
)
log
(
2
π
n
)
)
−
k
2
−
1
12
k
n
+
k
4
−
1
360
k
3
n
3
−
k
6
−
1
1260
k
5
n
5
+
O
(
1
n
6
)
{\displaystyle \log {\binom {kn}{n,n,\cdots ,n}}=kn\log(k )+{\frac {1}{2}}\left(\log(k )-(k-1)\log(2\pi n)\right)-{\frac {k^{2}-1}{12kn}}+{\frac {k^{4}-1}{360k^{3}n^{3}}}-{\frac {k^{6}-1}{1260k^{5}n^{5}}}+O\left({\frac {1}{n^{6}}}\right)}
so for example,
(
2
n
n
)
∼
2
2
n
n
π
{\displaystyle {\binom {2n}{n}}\sim {\frac {2^{2n}}{\sqrt {n\pi }}}}
Interpretations
edit
Ways to put objects into bins
edit
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k 1 objects in the first bin, k 2 objects in the second bin, and so on.[1 ]
Number of ways to select according to a distribution
edit
In statistical mechanics and combinatorics , if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution {n i } on a set of N total items, n i represents the number of items to be given the label i . (In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by
Choosing n 1 of the total N to be labeled 1. This can be done
(
N
n
1
)
{\displaystyle {\tbinom {N}{n_{1}}}}
ways.
From the remaining N − n 1 items choose n 2 to label 2. This can be done
(
N
−
n
1
n
2
)
{\displaystyle {\tbinom {N-n_{1}}{n_{2}}}}
ways.
From the remaining N − n 1 − n 2 items choose n 3 to label 3. Again, this can be done
(
N
−
n
1
−
n
2
n
3
)
{\displaystyle {\tbinom {N-n_{1}-n_{2}}{n_{3}}}}
ways.
Multiplying the number of choices at each step results in:
(
N
n
1
)
(
N
−
n
1
n
2
)
(
N
−
n
1
−
n
2
n
3
)
⋯
=
N
!
(
N
−
n
1
)
!
n
1
!
⋅
(
N
−
n
1
)
!
(
N
−
n
1
−
n
2
)
!
n
2
!
⋅
(
N
−
n
1
−
n
2
)
!
(
N
−
n
1
−
n
2
−
n
3
)
!
n
3
!
⋯
.
{\displaystyle {N \choose n_{1}}{N-n_{1} \choose n_{2}}{N-n_{1}-n_{2} \choose n_{3}}\cdots ={\frac {N!}{(N-n_{1})!n_{1}!}}\cdot {\frac {(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}}\cdot {\frac {(N-n_{1}-n_{2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}}\cdots .}
Cancellation results in the formula given above.
Number of unique permutations of words
edit
Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI.
The multinomial coefficient
(
n
k
1
,
…
,
k
m
)
{\displaystyle {\binom {n}{k_{1},\ldots ,k_{m}}}}
is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps, is
(
11
1
,
4
,
4
,
2
)
=
11
!
1
!
4
!
4
!
2
!
=
34650.
{\displaystyle {11 \choose 1,4,4,2}={\frac {11!}{1!\,4!\,4!\,2!}}=34650.}
Generalized Pascal's triangle
edit
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex . This provides a quick way to generate a lookup table for multinomial coefficients.
See also
edit
References
edit
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Multinomial_theorem&oldid=1218869408 "
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