The Multinomial Theorem
by aoum, Mar 22, 2025, 11:16 PM
The Multinomial Theorem: A Generalization of the Binomial Theorem
The Multinomial Theorem is a powerful generalization of the Binomial Theorem. While the Binomial Theorem describes the expansion of powers of a binomial expression, the Multinomial Theorem extends this concept to powers of a polynomial with multiple terms. This theorem is fundamental in combinatorics, algebra, and probability theory.
1. Statement of the Multinomial Theorem
For any positive integer
and for any nonnegative integer partition 
such that:
![\[
k_1 + k_2 + \dots + k_m = n,
\]](//latex.artofproblemsolving.com/e/f/a/efa1c721e624e90db4ddd8eaa41c457c2b8d2daf.png)
Theth power of the sum of 
variables 
can be expanded as:
![\[
(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n} \binom{n}{k_1, k_2, \dots, k_m} x_1^{k_1} x_2^{k_2} \dots x_m^{k_m},
\]](//latex.artofproblemsolving.com/6/a/b/6aba237afb2fa0cd3df7c92d74072b219dfd55b2.png)
Where:
![\[
\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1! k_2! \dots k_m!}
\]](//latex.artofproblemsolving.com/d/2/0/d20d5aebd31fba91f3dbd8a1ff9e10c0a4829662.png)
is the multinomial coefficient, which counts the number of ways to partition objects into groups of sizes .
2. Understanding the Multinomial Coefficient
The multinomial coefficient:
represents the number of ways to arrange items where there are 
of one type, 
of a second type, and so on. It generalizes the binomial coefficient, which corresponds to the case 
.
For example:
![\[
\binom{5}{2, 2, 1} = \frac{5!}{2!2!1!} = \frac{120}{4} = 30,
\]](//latex.artofproblemsolving.com/1/8/4/184ff2243a625ce7109d971ed92ae3c5319f99db.png)
which means there are 30 ways to arrange 2 objects of type 1, 2 objects of type 2, and 1 object of type 3.
3. Examples of the Multinomial Expansion
4. Proof of the Multinomial Theorem
We prove the Multinomial Theorem using combinatorial reasoning:
Consider the expansion of:
![\[
(x_1 + x_2 + \dots + x_m)^n,
\]](//latex.artofproblemsolving.com/5/1/b/51bb9ffe1071a381b2571aae43c06c68d7505a83.png)
When multiplying the factors, each term in the expansion is the product of choosing one 
from each factor. To form a term:
![\[
x_1^{k_1} x_2^{k_2} \dots x_m^{k_m},
\]](//latex.artofproblemsolving.com/8/1/8/8180ba3f88dc5d9680c1c2351d253b5ae81d59cb.png)
we must choose of the 
terms, of the 
terms, and so on. The number of ways to do this corresponds to the number of ways to partition items into groups of sizes , which is given by:
![\[
\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1! k_2! \dots k_m!}.
\]](//latex.artofproblemsolving.com/e/3/d/e3d211095b86e7cee2c2a568e1d547ba34610b7e.png)
Summing over all valid partitions gives the full expansion.
5. Special Cases of the Multinomial Theorem
6. Applications of the Multinomial Theorem
7. Example Problems
8. Conclusion
The Multinomial Theorem is a natural extension of the Binomial Theorem that describes the expansion of powers of a polynomial with multiple variables. Its applications are vast, ranging from combinatorics and algebra to probability and statistics. Understanding the theorem and its proof provides deep insight into the structure of polynomial expansions and the combinatorial nature of coefficients.
9. References
The Multinomial Theorem is a powerful generalization of the Binomial Theorem. While the Binomial Theorem describes the expansion of powers of a binomial expression, the Multinomial Theorem extends this concept to powers of a polynomial with multiple terms. This theorem is fundamental in combinatorics, algebra, and probability theory.
Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI.
1. Statement of the Multinomial Theorem
For any positive integer
and for any nonnegative integer partition 
such that:![\[
k_1 + k_2 + \dots + k_m = n,
\]](http://latex.artofproblemsolving.com/e/f/a/efa1c721e624e90db4ddd8eaa41c457c2b8d2daf.png)
The
th power of the sum of 
variables 
can be expanded as:![\[
(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n} \binom{n}{k_1, k_2, \dots, k_m} x_1^{k_1} x_2^{k_2} \dots x_m^{k_m},
\]](http://latex.artofproblemsolving.com/6/a/b/6aba237afb2fa0cd3df7c92d74072b219dfd55b2.png)
Where:
![\[
\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1! k_2! \dots k_m!}
\]](http://latex.artofproblemsolving.com/d/2/0/d20d5aebd31fba91f3dbd8a1ff9e10c0a4829662.png)
is the multinomial coefficient, which counts the number of ways to partition
objects into groups of sizes .2. Understanding the Multinomial Coefficient
The multinomial coefficient:
represents the number of ways to arrange
items where there are 
of one type, 
of a second type, and so on. It generalizes the binomial coefficient, which corresponds to the case 
.For example:
![\[
\binom{5}{2, 2, 1} = \frac{5!}{2!2!1!} = \frac{120}{4} = 30,
\]](http://latex.artofproblemsolving.com/1/8/4/184ff2243a625ce7109d971ed92ae3c5319f99db.png)
which means there are 30 ways to arrange 2 objects of type 1, 2 objects of type 2, and 1 object of type 3.
3. Examples of the Multinomial Expansion
- Example 1: Expand
.
We list all possible partitions of 3:
![\[
\begin{aligned}
&k_1 + k_2 + k_3 = 3: \\
&(3, 0, 0), (2, 1, 0), (2, 0, 1), (1, 2, 0), (1, 1, 1), (1, 0, 2), (0, 3, 0), (0, 2, 1), (0, 1, 2), (0, 0, 3).
\end{aligned}
\]](//latex.artofproblemsolving.com/1/8/2/182b9f6094e31c01200215cfa8617f6fc0fbea21.png)
Applying the Multinomial Theorem:
![\[
(x + y + z)^3 = \sum \binom{3}{k_1, k_2, k_3} x^{k_1} y^{k_2} z^{k_3},
\]](//latex.artofproblemsolving.com/2/5/4/2548358650149d65848e862de2b1a0e5346223be.png)
which expands to:
![\[
x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3.
\]](//latex.artofproblemsolving.com/8/d/0/8d0de7e68d0a93adf02e049077f5cb1cb3e00c87.png)
- Example 2: Find the coefficient of
in the expansion of 
.
We identify
, 
, and 
, so:
![\[
\binom{9}{2, 3, 4} = \frac{9!}{2!3!4!} = \frac{362880}{48} = 7560.
\]](//latex.artofproblemsolving.com/9/f/7/9f741535beb3a15d0d1f1f4b8c117f3ae0fd2e1f.png)
Thus, the coefficient is 7560.
.![\[
\begin{aligned}
&k_1 + k_2 + k_3 = 3: \\
&(3, 0, 0), (2, 1, 0), (2, 0, 1), (1, 2, 0), (1, 1, 1), (1, 0, 2), (0, 3, 0), (0, 2, 1), (0, 1, 2), (0, 0, 3).
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/8/2/182b9f6094e31c01200215cfa8617f6fc0fbea21.png)
![\[
(x + y + z)^3 = \sum \binom{3}{k_1, k_2, k_3} x^{k_1} y^{k_2} z^{k_3},
\]](http://latex.artofproblemsolving.com/2/5/4/2548358650149d65848e862de2b1a0e5346223be.png)
![\[
x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3.
\]](http://latex.artofproblemsolving.com/8/d/0/8d0de7e68d0a93adf02e049077f5cb1cb3e00c87.png)
in the expansion of 
.
,
,
,![\[
\binom{9}{2, 3, 4} = \frac{9!}{2!3!4!} = \frac{362880}{48} = 7560.
\]](http://latex.artofproblemsolving.com/9/f/7/9f741535beb3a15d0d1f1f4b8c117f3ae0fd2e1f.png)
4. Proof of the Multinomial Theorem
We prove the Multinomial Theorem using combinatorial reasoning:
Consider the expansion of:
![\[
(x_1 + x_2 + \dots + x_m)^n,
\]](http://latex.artofproblemsolving.com/5/1/b/51bb9ffe1071a381b2571aae43c06c68d7505a83.png)
When multiplying the
factors, each term in the expansion is the product of choosing one 
from each factor. To form a term:![\[
x_1^{k_1} x_2^{k_2} \dots x_m^{k_m},
\]](http://latex.artofproblemsolving.com/8/1/8/8180ba3f88dc5d9680c1c2351d253b5ae81d59cb.png)
we must choose
of the 
terms, of the 
terms, and so on. The number of ways to do this corresponds to the number of ways to partition items into groups of sizes , which is given by:![\[
\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1! k_2! \dots k_m!}.
\]](http://latex.artofproblemsolving.com/e/3/d/e3d211095b86e7cee2c2a568e1d547ba34610b7e.png)
Summing over all valid partitions gives the full expansion.
5. Special Cases of the Multinomial Theorem
- When , the Multinomial Theorem reduces to the Binomial Theorem:
![\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n - k} y^k.
\]](//latex.artofproblemsolving.com/5/0/2/502d88feb9e689328d66f0d39f13b8cadf006ab3.png)
- When
, the Multinomial Theorem gives the trinomial expansion:
![\[
(x + y + z)^n = \sum_{k_1 + k_2 + k_3 = n} \binom{n}{k_1, k_2, k_3} x^{k_1} y^{k_2} z^{k_3}.
\]](//latex.artofproblemsolving.com/d/0/9/d093a564f8b0aeac8b0d78e989a6e3717f6dadd6.png)
![\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n - k} y^k.
\]](http://latex.artofproblemsolving.com/5/0/2/502d88feb9e689328d66f0d39f13b8cadf006ab3.png)
,![\[
(x + y + z)^n = \sum_{k_1 + k_2 + k_3 = n} \binom{n}{k_1, k_2, k_3} x^{k_1} y^{k_2} z^{k_3}.
\]](http://latex.artofproblemsolving.com/d/0/9/d093a564f8b0aeac8b0d78e989a6e3717f6dadd6.png)
6. Applications of the Multinomial Theorem
- Combinatorics: Counting arrangements and partitions of sets.
- Probability: Analyzing multinomial distributions in statistics.
- Algebra: Expanding powers of polynomials.
- Physics: Modeling outcomes in multi-particle systems.
7. Example Problems
- Example 1: How many ways can you distribute 10 identical objects into 4 distinct bins?
- Example 2: Find the coefficient of
in the expansion of 
.
![\[
\binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = \boxed{286}.
\]](http://latex.artofproblemsolving.com/5/f/9/5f93986d588c6adf1c334086fd3c2093da3f01c8.png)
in the expansion of 
.
,![\[
\binom{10}{2, 3, 5} = \frac{10!}{2!3!5!} = \boxed{2520}.
\]](http://latex.artofproblemsolving.com/6/6/a/66a5b9119143fdcfe59e6dbcfa38ca46087ac6cf.png)
8. Conclusion
The Multinomial Theorem is a natural extension of the Binomial Theorem that describes the expansion of powers of a polynomial with multiple variables. Its applications are vast, ranging from combinatorics and algebra to probability and statistics. Understanding the theorem and its proof provides deep insight into the structure of polynomial expansions and the combinatorial nature of coefficients.
9. References
- Wikipedia: Multinomial Theorem
- Concrete Mathematics, Graham, Knuth, and Patashnik.
March 2025