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

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3
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1answer
30 views

Combinatorics: How do you find the coefficient in the given expression?

The question asks me to find the coefficient of the term $x^6y^4$ in the expression $(xy^2+x^2+3y)^7$. This was pretty simple. This is how I did it: $$(xy^2+x^2+3y)^7 = \sum_{a+b+c = n} (xy^2)^a + ...
1
vote
2answers
39 views

Multinomial coefficient

$n_{1},...,n_{m}$ are nonnegative integers $n = n_{1}+...+n_{m}$ $W[n_{1},...,n_{m}]$ denotes the # of ways to place $n$ balls (of labels $1, 2, . . . , n$) into $m$ boxes $B_{1},...,B_{m}$ such ...
0
votes
3answers
69 views

Help with sum of coefficients please!

Problem 1: "Imagine that the polynomial $(1 + x - y)^3$ is converted to the standard form. What is the sum of its coefficients?" Problem 2 (continued): "What is the sum of the coefficients of the ...
2
votes
0answers
33 views

Proof of hockey stick like theorem for multinomial coefficients

I discovered following identity via a programming exercise. Looking for the way how it prove it by induction or some other way. $$\binom{a_1+a_2+\cdots+a_t}{a_1,a_2,\cdots,a_t}=1+\sum_{i=2}^t ...
0
votes
1answer
25 views

Notation for writing multinomial coefficient as sum of smaller multinomial coefficients

This question is an attempt to extend the Pascal triangle's hockey stick identity to multinomial coefficients as asked in question Hockey-Stick Theorem for Multinomial Coefficients. Consider the ...
1
vote
1answer
32 views

Hockey-Stick Theorem for Multinomial Coefficients

Pascal's triangle has this famous hockey stick identity. $$ \binom{n+k+1}{k}=\sum_{j=0}^k \binom{n+j}{j}$$ Wonder what would be the form for multinomial coefficients?
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2answers
55 views

Inequality proof involving multinomial coefficients

How may I proceed to prove/disprove following inequality? $$\frac{n^n}{p_1^{p_1}\cdot p_2^{p_2}\cdots p_k^{p_k}}>\frac{n!}{p_1! p_2!\cdots p_k!} $$ where $\sum_{i=1}^k p_i=n$ It seems, using ...
0
votes
1answer
46 views

Sum of the first k multi-nomial coefficients for fixed n

Multinomial coefficients are define as $$\binom{n}{k_1,k_2,\cdots,k_m}$$ where $n=k_1+k_2+\cdots+k_m$ Is there closed form solution available for sum of first $t$ multinoial coefficients? Order of ...
0
votes
1answer
50 views

The sum which gives $3^n$

So I have the following which I must prove : $$\sum_{(n_1,n_2,n_3)\,:\,n_1+n_2+n_3=n} \binom{n}{n_1, n_2, n_3} = 3^n$$ I'm not sure where I must begin. This is a multinomial.
2
votes
1answer
35 views

Confused about multinomials. Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$?

Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$? The definition for multinomial says $a+b+c=n$ must hold or else $\binom{n}{a,b,c}=0$. I found that if ...
0
votes
0answers
33 views

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 ...
0
votes
2answers
110 views

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 ...
0
votes
0answers
35 views

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 ...
-1
votes
1answer
38 views

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 ...
-5
votes
2answers
79 views

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 ...
0
votes
0answers
31 views

How can I easily find the coefficient of an arbitrary monomial an expansion?

With the binomial theorem and multinomial theorems you can easily find the coefficient of monomials in expansions of expressions like $(x+y+z)^6$, but what if I have a more complex expression like ...
1
vote
2answers
14 views

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: ...
1
vote
1answer
63 views

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 $k_1, k_2, \ldots , k_6$ ...
3
votes
1answer
32 views

General solution to expressions, without calculating exact roots (A generalization of Newton's identities)

Consider the following equations: $$A_1^1=\sum_iy_i=y_1+y_2+\ldots+y_m=a_1$$ $$A_2^1=\sum_{i_1,i_2}y_{i_1}y_{i_2}=a_2\,\,,i_1< i_2$$ $$A_3^1=\sum_{i_1,i_2,i_3}y_{i_1}y_{i_2}y_{i_3}=a_3\,\,,i_1< ...
4
votes
2answers
90 views

A meaningful sum of multinomials

Consider paths that touch $n$ nodes of a complete graph, and let's number these nodes from $1$ to $n$. The number of paths that pass $m_1$ times through node $1$, $m_2$ times through node $2$, etc., ...
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votes
2answers
60 views

Distribution of identical objects among people

How to find the number of ways in which n identical objects can be divided among r persons where each person gets a maximum of k objects?
0
votes
1answer
74 views

How do I find the number of solutions of the equation $r_1 + r_2 + … + r_k = n$

I was studying the multinomial theorem: $(u_1+u_1+...u_k)^n=\sum\limits_{r_1+r_2+...r_k=n}\dfrac{n!}{r_1!r_2!...r_k!}u_1^{r_1}u_2^{r_2}...u_k^{r_k}$ and my book said that the number of terms in the ...
1
vote
0answers
48 views

Coefficient of expansion of $\,x\,$

What is the simplest way to find the coefficient of, for example, $\,x^{ 6 }$ in $\left(x+1\right)\cdot\left(x+2\right)\cdot \ldots\cdot\left(x+10\right)\,$? My teacher says that the easiest way is ...
0
votes
1answer
28 views

formula for multinomial expansion raised to three

who could kindly give me the formula for $$(x_1+x_2 + \cdots+ x_n) ^3,$$ in the form like the case $$(x_1+x_2 + \cdots+ x_n) ^2 = \sum^n_{i=1} x_i^2 + 2\sum_{1\leq i<j\leq n} x_ix_j.$$ Thanks
0
votes
0answers
51 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
10
votes
1answer
1k views

Why isnt 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 ...
2
votes
0answers
116 views

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:: ...
4
votes
2answers
147 views

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 ...
0
votes
2answers
112 views

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 ...
3
votes
1answer
63 views

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, ...
1
vote
2answers
70 views

Sum of terms in a multinomial expansion? (that is all coefficients are equal to one) [closed]

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.
1
vote
1answer
25 views

Is there a way to find expansion of powers of multinomials without any coefficients?

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 + ...
4
votes
2answers
68 views

Rolling two dice… [closed]

Let $A_n$ be the number of fives, $B_n$ the number of sixes and $C_n$ the number of eights in $n$ rolls of two dices. For which n do we have: $E(A_n) < E(min(B_n,C_n))$ ?
1
vote
1answer
104 views

Sum of variances of multinomial distribution.

I've k fair coins, and I would like to know the number of heads obtained in $n$ trials. But that is simple binomial distribution. But if I want to find out how much it varies from binomial ...
1
vote
0answers
120 views

Multinomial Theorem for Negative Exponents

Using an analog to Newton's binomial theorem with negative exponents, is it true that $$ \begin{align} \left(\sum_{k=0}^mx^k\right)^{-n} & = \sum_{0\le ...
0
votes
1answer
29 views

Numerically stable calculation of multinomial probabilities

I'm looking for a numerically stable method to compute expressions of the form $$\frac{(a+b+c+d)!}{a!b!c!d!}\left(\frac{1}{4}\right)^{a+b+c+d}$$ So far I've been using a compensated sum algorithm to ...
1
vote
1answer
107 views

Probability Generating Function of a Negative Multinomial Distribution

Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability ...
1
vote
1answer
145 views

I need help in calculating the sum of the coefficients of even powers of $x$ in $(1+x-2x^2)^6$

I need to calculate the sum of coefficients of even powers of $x$ in $$(1+x-2x^2)^6$$ I don't know much about the multinomial theorem, but i know the basics pretty well. I have some ideas of solving ...
1
vote
0answers
38 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
3
votes
2answers
179 views

Find the coefficient of $x^4$ in the expansion of $(1 + 3x + 2x^3)^{12}$?

I have not learnt the multinomial theorem yet, and was trying to approach this using the binomial theorem. I divided the terms as $a$ being $(1+3x)$ and $b$ being $2x^3$. I then used $${12\choose ...
4
votes
3answers
359 views

Sum of coefficients in an multinomial expression.

Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is: What I tried to do was make pairs of two terms and the expand it as a binomial ...
0
votes
1answer
58 views

Ways of selecting at most n objects from a set containing k distinct elements where each element can occur any number of times.

I have a box with a maximum capacity of n elements. A state of the box is defined by the elements in it. There is an infinitely large heap which has k distinct elements; each element is available ...
0
votes
1answer
91 views

Upper bound to multinomial coefficient sum

I'm currently stuck on what seems like a very trivial problem. I have the following calculation $$ \sum_{k_1+k_2=0}^{n} {n \choose n - k_1 - k_2, k_1, k_2}^2 \le \sum_{k_1+k_2=0}^{n} {n \choose ...
3
votes
1answer
84 views

$(1+x+x^2)^{1061}=a_0+a_1+…$ then what is the value of $(1-a_1^{2}+a_2^{2}…)$ in terms of $a_n$

$(1+x+x^2)^{1061}=a_0+a_1x+...+a_{2122}x^{2122}$ then what is the value of $(1-a_1^{2}+a_2^{2}-a_3^{2}...)$ in terms of single $a_n$ ? n lies between 0 and 2122 . how to get in terms on $a_n$ ? ...
5
votes
1answer
160 views

Complex Analysis proof of multinomial expression

I've recently come across the following identity $$ \displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
1
vote
1answer
90 views

Multinomial Expansion-Example

Is coefficient of $x^{20}$ in $(1-x+x^2)^{20}$ and $(1+x-x^2)^{20}$ same? Can someone tell me how should i apply multinomial theorem to this problem?
1
vote
1answer
66 views

Maximum of trinomial coefficient

I read a proof about the simple random walk in 3 dimensions and couldn't understand the following statement: $$\frac{n!}{k!j!(n-k-j)!}$$ has the maximum when $\ k, j $ and $\ n-k-j$ are as close to ...
0
votes
1answer
44 views

Find multiplier of x^6 in (a + bx + cx^2)^4 (Multinomail theorem)

Have a simple problem where I need to find multiplier of $x^6$ in $(2+x+3*x^2)^4$ I tried doing it like this: $ a = 2; b = x; c = 3*x^2$ Using multinomial theorem : Possible combinations that give ...
0
votes
1answer
26 views

Cofficient of x in a product

How can I efficiently find the coefficient of $x^m$ in the following product - $\prod\limits_{i=1}^{n-1}(1 - p_i + p_ix)$
1
vote
1answer
39 views

Is there a straight forward way to count the number of monomials in a multinomial equation?

The multinomial theorem states that: $$(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n}{n \choose {k_1, k_2, \dots, k_m}}\prod_{1 \le t \le m} {x_t}^{k_t}$$ The number of monomials ...