| Version 4 |
Version 3 |
| A multinomial is a mathematica expression consisting of two or more terms, e.g. |
A multinomial is an expression of the form $a_1 x_1 + a_2 x_2 + \ldots + a_k x_k$. |
| $$a_1 x_1 + a_2 x_2 + \ldots + a_k x_k.$$ |
The multinomial theorem provides the general form of the expansion of powers of this |
| The multinomial theorem provides the general form of the expansion of the powers of this |
|
| expression, in the process specifying the multinomial coefficients which are found in that expansion. The expansion is: |
expression, in the process specifying the multinomial coefficients which are found in that expansion. The expansion is: |
| \begin{equation} |
\begin{equation} |
| (x_1 + x_2 + \ldots + x_k)^n &=& |
(x_1 + x_2 + \ldots + x_k)^n &=& |
| \Sigma \frac{n!}{n_1! n_2! \cdots n_k!} x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k} |
\Sigma \frac{n!}{n_1! n_2! \cdots n_k!} x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k} |
| \end{equation} |
\end{equation} |
| where the sum is taken over all multi-indices $(n_1, \ldots n_k)\in\mathbb{N}^k$ that |
where the sum is taken over all multi-indices $(n_1, \ldots n_k)\in\mathbb{N}^k$ that |
| sum to $n$. |
sum to $n$. |
