# Tagged Questions

For questions related to multinomial coefficients, a generalization of binomial coefficients.

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### Best coefficient between two data sets

I want to determinate a sensor coefficient but I struggle with a basic math problem... Here are my values : ...
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### What's the formula to map between multiindices and indices?

What is the formula to map between multiindices and indices? By multiindex, I mean a variable $I\in\mathbb{N}^d$ where $|I|=\sum\limits_{i=1}^d I_i=n$. Here, $d$ denotes the dimension. Basically, ...
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### Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
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### Is my proof to proove that $\frac{n!}{p_1!p_2!p_3!…p_m!}\in \mathbb{N}$ valid?

I wish to prove that $\frac{n!}{p_1!p_2!p_3!....p_m!}$ is an integer, were, $p_1+p_2+p_3+...+p_m=n$ and $p_i, n\in \mathbb{N}$. Pleace do check the validity of my proof. Let us consider the following ...
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### The expansion of $(a+b+c+d)^{20}$ [closed]

Let us consider the expansion of $$(a+b+c+d)^{20}.$$ Find: The coefficients of $a^{11}b^6c^2d$ and $a^{11}b^9$, The total number of terms of this expansion, The sum of all the coefficients. Thank ...
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### Format of Binomial Coefficient vs Multinomial Coefficient

It would not be unreasonable to assume that a special case of the multinomial coefficient is the binomial coefficient where there are only two terms. Why is the binomial coefficient written ...
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### The number of odd coefficients in the expansion of $(x^2+x+1)^n$

Find the number of odd coefficients in terms of $n$, in the expansion of $(x^2+x+1)^n$ where $n$ is a positive integer. I have tried directly applying multinomial and condition for it to be odd by ...
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### Another Generalization of Chu–Vandermonde Identity $\sum_{|\alpha|=n}{n\choose\alpha}^2=?$

Chu-Vandermonde identity states $$\sum_{k=0}^n{{n\choose k}^2}=\sum_{k=0}^n{{n\choose k}{n\choose{n-k}}}={2n\choose n}$$ The proof is inspired by expanding $(1+x)^n(1+x)^n$ to two kinds of series and ...
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### Combinations, when placing n objects into k boxes, each box has its own size and the order in them doesn't matter?

I have n objects and k boxes, each box has its own size. The arrangement of objects in a box doesn't matter. How many combinations are there, respectively the formula? Say we have 7 objects and 3 ...
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### Pascal Triangle Ball Conundrum [closed]

Imagine we have a Pascal Triangle Pin Board: A ball is dropped, and at every pin it has an equal chance of falling left or right. If we drop $32$ balls, what is the probability that the final ...
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### Why is the maximum sum of two proportions = 1 with the multinomial logit

Suppose I have two numbers, actually two proportions, a and b, where: ...
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### coefficient on $s^{14}$ in generating function

I have $(s+s^2+s^3+s^4+s^5+s^6)^7$, and I'm trying to find the coefficient on $s^{14}$. I've tried using the multinomial theorem, but that leads to the problem of finding all $k_1, k_2, \ldots , k_6$ ...
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### Why isn't there only one way of painting these horses?

If you have $11$ identical horses in how many ways can you paint 5 of them red 3 of them blue and 3 brown? My intuition instantly tells me there is only one way of doing this. I mean if the horses ...
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### Expected Power Product of rolling a dice .

A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears. How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$ My attempts:: ...
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### A summation involving multinomial coefficient

We need to find out $$\sum {\binom{N}{a_1,a_2,a_3...a_B} a_1^{\alpha}a_2^{\alpha}...a_C^{\alpha} }$$ $$a_1+a_2...a_B=N, \alpha>0 ,0\lt C \le B$$ All are nonnegative integers. We need to sum ...
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### Multinomial Coefficients Definition in expansion of $(1+x+x^2+\cdots+x^l)^n$

The literature defines multinomial coefficients (or extended bnomial coefficients) as $$\binom{n}{r_1,r_2,\cdots,r_l} = \frac{n!}{r_1!r_2!\cdots r_l!}$$ where $$r_1+r_2+\cdots+r_l = n$$ Which is ...
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### Integer Partitions and distinguishable permutations

I'm not a mathematician but I'm faced with a problem where I can't find an answer, also because I do not know what I shall ask for: I have to deal with partitions of an integer k, only small values, ...
How to sum the series $a^3+b^3+c^3+a^2b+a^2c+b^2a+b^2c+c^2a+c^2b+abc$? And in general for any multinomial expansion.
For example, $(a + b + c)^3 = a^3 + b^3 + c^3 + 3ab^2 + 3ac^2 + 3a^2b + 3a^2c + 3bc^2 + 3b^2c + 6abc$ Knowing the value of a, b and c, is there a way to find this without the coefficients i.e. \$a^3 + ...