Ask Question
An extension to the binomial theorem. It gives the expansion of a multinomial $(x_0,\dots,x_{m-1})^n$.
154 questions- Bountied 0
- Unanswered
- Frequent
- Score
- Unanswered (my tags)
multinomial formular and power series connection
Let $f(\boldsymbol{\rm{x}} \mid \boldsymbol{n})=\frac{\left(x_{0}+n_{1}+\cdots+n_{m}-1\right)!}{\left(x_{0}-1\right)!{n_1!} \ \cdots \ {n_m!}}y^{x_{0}} \prod_{i=1}^{m}{x_{i}^{n_i}}$ where $\mathbf{x} =... sequences-and-series multinomial-theorem- 1
Multinomial theorem with multivariate terms?
Let $S=\{a,b,c,d,...\}$. Let $P_n=(abc+abd+acd+...+ab+ac+ad+...a+b+c+d...)^n$. In addition, there's the condition that for all variables, $x^n=x$ (maybe it'll be easier without this?). Is there ... multivariable-calculus multinomial-theorem- 51
Multinomial Theorem with "collaspable" variables
Is there something similar to the multinomial theorem but with the added wrinkle that any variable raised to any positive integer power is itself? multinomial-theorem- 51
Find the Coefficient of $x^8$ by Multinomial Theorem Expansion for $(1+x^2-x^3)^9$
Problem: Find the coefficient of the term $x^8$ for the expansion of $(1+x^2-x^3)^9$ Attempt: By the multinomial theorem: $$(1+x^2-x^3)^9=\sum_{b_1+b_2+b_3=9}{9\choose b_1,b_2,b_3}(1)^{b_1}(x^2)^{b_2}(... multinomial-coefficients multinomial-theorem- 178
An equivalent formula for $\sum_{1\le i_1\lt i_2 \dots \lt i_n\le n} a_{i_1} a_{i_2} \dots a_{i_n}$
I know that the following holds: $\sum_{1\le i\lt j\le n} a_j a_k = \frac12\left(\left(\sum a_i\right)^2-\sum a_i^2\right)$ The question is: does some equivalent formula holds for products of $q>... combinatorics binomial-theorem multinomial-theorem- 21
How to find the $r$-th term of a trinomial expansion?
For a binomial expansion $(x^a + y^b)^n$ finding the rth term simply meant that the exponent of $y$ should be $b$ times $n$. How about for a multinomial expansion, $(x^a + y^b + z^c)^n$? I see in some ... multinomial-theorem- 271
Find the $x^n$ coefficient of $(1+x+x^2)^n$
I've tried a bunch of different groupings of the three terms so that I could use the binomial expansion forumula, but I haven't been able to go much further than that. This is an example of what I've ... combinatorics binomial-coefficients generating-functions multinomial-coefficients multinomial-theorem- 89
Issues understanding the multinomial theorem and its multiindex notation
$$(x_1+x_2+...+x_m)^n=\sum_{(k_1 + k_2 +... +k_m) \ = \ n} {n \choose k_1,k_2...k_m} \prod^m_{t=1}x_t^{k_t}$$ Let's do $(a+b+c)^3$. That means $a =x_1, b =x_2, c=x_3=x_m$. The multiindex below the ... combinatorics summation products index-notation multinomial-theorem- 848
How to get multinomial sum without coefficients
We know that a multinomial sum is given by: $(x_1 + x_2 + x_3 + ... + x_m)^n = \sum\limits_{k_1+k_2+k_3+...+k_m=n} {n \choose {k_1, k_2, k_3, ... k_m}} \prod\limits_{t=1}^m x_t^{k_t}$ I want to ... sequences-and-series multinomial-theorem- 359
Number of terms in product of two monomials with common terms
I am trying to find the number of terms in the expression $$(x+y+z)^{20}(w+x+y+z)^2$$. I understand that the number of terms in $(x+y+z)^{20} = \binom{22}{2}$ and the number of terms in $(w+x+y+z)^2 = ... combinatorics binomial-theorem multinomial-theorem- 73
Is it possible to determine the number of terms in a multinomial expansion, if all terms are exponents of $x$?
Is it possible to determine the number of terms in a multinomial expansion, if all terms are exponents of $x$? For example, number of terms in the expansion of $(1+x^2+x^4+x^5)^7$ ? Clearly, the ... combinatorics binomial-theorem multinomial-theorem- 4,040
Coefficient of $1$ in the expansion of $\left(1+x+\frac{1}{x}\right)^n$
What is the coefficient of $1$ in the expansion of $(1+x+\frac{1}{x})^n$? In other words, what is the sum of the coefficients of $x^ky^k$ in the expansion of $(1+x+y)^n$? Here, $n$ is a positive ... asymptotics binomial-coefficients multinomial-theorem- 499
Question about multinomial expansion
The teacher briefly glossed over the multinomial theorem and then dropped this seemingly monstrous homework problem on us: Find the coefficient of $x^{12}$ in the expansion of: $(x^5+x^6+x^7+\ldots )^... combinatorics multinomial-coefficients multinomial-theorem- 3
No. of integral solutions of an equation with upper and lower bound without 'generating the function' method
Question from my book Now, I know how to solve this using 'generating the function' method. But I cant figure out what method is used in the solution of the problem given in the book,especially from ... permutations combinations multinomial-theorem- 17
How to show $\begin{pmatrix}1&1 \\ 1&1\end{pmatrix}^n = 2^{n-1}\begin{pmatrix}1&1 \\ 1&1\end{pmatrix}$?
Initial note: I'm interested in the combinatorics aspect of the following problem, not how to proof the relation in general. The idea is to show the following relationship: $$ \begin{pmatrix}1&1 \... combinatorics matrices multinomial-coefficients multinomial-theorem- 237
15 30 50 per page12345…11 Next
