Multinomial theorem
Multinomial Theorem
The Multinomial Theorem is a generalization of the binomial theorem to polynomials with any number of terms. While the binomial theorem deals with the expansion of expressions of the form $(x + y)^n$, the multinomial theorem deals with expressions that involve more than two terms, such as $(x_1 + x_2 + \ldots + x_m)^n$.
The Multinomial Coefficient
Before diving into the theorem itself, it is important to understand the concept of a multinomial coefficient. The multinomial coefficient is a generalization of the binomial coefficient and is defined for a given non-negative integer $n$ and a sequence of non-negative integers $k_1, k_2, \ldots, k_m$ such that $k_1 + k_2 + \ldots + k_m = n$. It is denoted and calculated as follows:
$$ \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! k_2! \ldots k_m!} $$
This coefficient represents the number of ways to partition a set of $n$ items into $m$ containers, with $k_i$ items in the $i$-th container.
The Multinomial Theorem
The multinomial theorem states that for any non-negative integer $n$ and any real or complex numbers $x_1, x_2, \ldots, x_m$, the expansion of $(x_1 + x_2 + \ldots + x_m)^n$ is given by:
$$ (x_1 + x_2 + \ldots + x_m)^n = \sum \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \ldots x_m^{k_m} $$
where the sum is taken over all sequences of non-negative integers $k_1, k_2, \ldots, k_m$ such that $k_1 + k_2 + \ldots + k_m = n$.
Differences and Important Points
Here is a table comparing the binomial and multinomial theorems:
| Feature | Binomial Theorem | Multinomial Theorem |
|---|---|---|
| Terms | 2 terms (x + y) | m terms $(x_1 + x_2 + \ldots + x_m)$ |
| Coefficients | Binomial coefficients $\binom{n}{k}$ | Multinomial coefficients $\binom{n}{k_1, k_2, \ldots, k_m}$ |
| Expansion | $(x + y)^n = \sum \binom{n}{k} x^k y^{n-k}$ | $(x_1 + x_2 + \ldots + x_m)^n = \sum \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \ldots x_m^{k_m}$ |
| Summation | Over all $k$ such that $0 \leq k \leq n$ | Over all sequences $(k_1, k_2, \ldots, k_m)$ such that $k_1 + k_2 + \ldots + k_m = n$ |
| Use Case | When dealing with two variables | When dealing with more than two variables |
Examples
Example 1: Expansion of a Trinomial
Let's expand $(a + b + c)^2$ using the multinomial theorem.
$$ (a + b + c)^2 = \sum \binom{2}{k_1, k_2, k_3} a^{k_1} b^{k_2} c^{k_3} $$
where $k_1 + k_2 + k_3 = 2$. The possible sequences of $(k_1, k_2, k_3)$ are $(2, 0, 0)$, $(0, 2, 0)$, $(0, 0, 2)$, $(1, 1, 0)$, $(1, 0, 1)$, and $(0, 1, 1)$. Substituting these into the formula gives us:
$$ (a + b + c)^2 = \binom{2}{2, 0, 0} a^2 + \binom{2}{0, 2, 0} b^2 + \binom{2}{0, 0, 2} c^2 + \binom{2}{1, 1, 0} ab + \binom{2}{1, 0, 1} ac + \binom{2}{0, 1, 1} bc $$
Calculating the coefficients:
$$ (a + b + c)^2 = 1 \cdot a^2 + 1 \cdot b^2 + 1 \cdot c^2 + 2 \cdot ab + 2 \cdot ac + 2 \cdot bc $$
So the expanded form is:
$$ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc $$
Example 2: Multinomial Coefficients
Calculate the multinomial coefficient $\binom{5}{1, 2, 2}$.
$$ \binom{5}{1, 2, 2} = \frac{5!}{1! \cdot 2! \cdot 2!} = \frac{120}{2 \cdot 2} = 30 $$
This coefficient represents the number of ways to divide 5 objects into three groups where one group has 1 object and the other two groups have 2 objects each.
The multinomial theorem is a powerful tool in combinatorics and algebra, and it is essential for understanding the distribution of terms in the expansion of polynomials with more than two variables. It also has applications in probability theory and other areas of mathematics.