Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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Ways to think about the binomial coefficient

Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial ...
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295 views

Sum involving integer compositions and binomial coefficients

I came across an interesting identity involving binomial coefficients. I'm not sure if I'm looking at the identity the wrong way but I am not aware if this identity is known and if there is an (easy) ...
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54 views

Combinatorics problem involving binomial coefficient

I found this interesting problem in a Romanian mathematical magazine while preparing for the USAMO. Let $k$ be a non-zero natural number. Determine $x,y,z \in \Bbb N$ such that $$\binom {z+k}{x+y} - \...
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Median poker hand (Texas Hold’em)

I try to find the median poker hand (Texas Hold’em). The following is given: 1) There are 52 cards 2) Assuming seven of them are chosen randomly 3) Create the best possibility with 5 of these 7 ...
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40 views

Binomial identity in a finite field

Suppose we have a prime $p$ and consider $\mathbb{F}_q$ where $q=p^s$ for some $s$. Fix a positive integer $m \geq 2$ and let $t \leq m-1$. Let $r$ be a positive integer such that $0 \leq r \leq q^t-1$...
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Computing a sum involving binomial coefficients

I am doing some (pretty heavy) computations, and I am stuck at a point that can be rephrased as follows: Let $m>n\ge0$ be two integers. Compute $$\sum_{k=0}^n\binom{n}{k}\frac{1}{m-k}x^{n-...
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19 views

Number of terms in multivariate polynomial

We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have: $1+x+y+xy+...
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36 views

Vandermonde-type convolution with geometric term

Is there a closed-form solution to the following sum? \begin{align*} f(r, s, n) = \sum_{k=0}^{n}c^k\binom{r}{k}\binom{s}{n-k} \end{align*} I know this corresponds to find the coefficient of $x^n$ of ...
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58 views

Evaluate the combination of $\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$

Can any one help me please to get the approximate result of this combination problem using asymptotic notation: $$ \sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j} $$ Thanks
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37 views

Experiment by Bernoulli process

I have a question. Assume I carry out an experiment by Bernoulli process. I repeat the tests until the number of successful outcomes exceed the number of unsuccessful outcomes by m. What will be the ...
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33 views

Logic - Binomial Theorem

I could use some assistance with understanding this problem. I understand that there are ${n}\choose{k}$ is a representation of ${n}\choose{k}$ ways to choose k elements from a set of n elements, ...
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35 views

prime numbers - need a help

Helow, There is a question about prime numbers. Supposed that I already answer the first section. I try to answer the second section, but if n $\neq$ $2^{k}$ (for some k from the natural numbers, ...
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25 views

On a theorem of Hensel

In the paper Binomial coefficients modulo prime powers, Andrew Granville state the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
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24 views

Multinomial coefficients modulo a prime

Let $p$ be a prime and let $m \geq 1$. Lucas' theorem implies that the binomial coefficient ${p^m-1 \choose k}$ is not divisible by $p$ for any $0 \leq k \leq p^m-1$. I wonder if something similar ...
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86 views

How to prove this equality about Eulerian numbers?

I want to prove the following equality where $A(k,m)$ is the Eulerian number : $$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$ I previously proved ...
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41 views

A theorem about binomial coefficient module prime

For any integer $r$ and prime $p$, there is a integer $n$ which $\binom{2n}{n}\equiv r \pmod{p}$. I tried Lucas's theorem, but I was stuck. Suppose $r\neq 0$, otherwise we can let $n=p$. Let $n=\...
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42 views

Ho can I find the sum of this sequence?

There is a sequence $$ F_n= \begin{cases} aF_{n-1}+q^{n-2}F_{n-2},& n \text{ is even}\\ bF_{n-1}+q^{n-2}F_{n-2},& n \text{ is odd} \end{cases} $$ with the initial conditions $F_0 = 0 ...
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25 views

Pascal triangle with non equiprobable events

As many of you will know when flipping n times a coin where each side is equally probable we can calculate the probability of getting x times heads with the triangle of pascal, that would be ${n \...
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42 views

For any given $k$, show that an integer $n$ can be represented as: $n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$

For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$. My ...
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39 views

Does the property ${n\choose r}={n\choose n-r}$ have a name?

Due to the relation between Pascal's Triangle and the choose function in probability theory, we can deduce that $${n\choose r}={n\choose n-r}$$ because Pascal's Triangle is symmetric. This can also be ...
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48 views

Problem with induction of binomial coefficiency

(Sorry for making up math language, I am roughly translating math terms here) This is part of some of the induction exercises in the book "Otto Forster: Analysis 1" (1.2): $$\binom{x+y}{n}=\sum_{k=0}...
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For what values of $n$ do more than two thirds of subsets of $[n]$ have size beteen $\frac{n}{3}$ and $\frac{2n}{3}$

My actual problem is slightly different. For what values of $n$ do we have: $2\sum_{i=1}^{\lfloor n/3 \rfloor}\binom{n}{i}\geq \sum\limits_{i=\lfloor n/3 \rfloor+1}^{\lceil n/2 \rceil-1} \binom{n}{i}...
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Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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28 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
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37 views

Very odd binomial coefficients

The number of odd binomial coefficients in each row of Pascal's triangle is always a power of two although their sum rarely is. One of these rare occasions occurs for numbers of the form $\,$$n = 2^m ...
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28 views

Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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39 views

Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ \...
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Recurrence relationships and a “weighted Pascal's triangle”

I was thinking about this problem a few days ago and in the process I came up with what I can best describe as a two-dimensional recurrence relationship. It seemed obvious to me that this was ...
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42 views

Prove the following result on binomial coefficients

If $(1+x)^n=^n\!\!C_0+^n\!\!C_1x+^n\!\!C_2x^2+\cdots+^n\!\!C_nx^n$, then show that $$(^n\!\!C_0-^n\!\!C_2+^n\!\!C_4-^n\!\!C_6+\cdots)^2+(^n\!C_1-^n\!\!C_3+^n\!\!C_5-\cdots)^2\\ =^n\!\!C_0+^n\!\!C_1+^n\...
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Simple binomial coefficients sum question

I know $\sum_\limits{i=0}^n \dbinom{n}{i}=2^n$, and some others, but is there a simple expression when the lower and upper limits are arbitrary, i.e.; $$\sum_\limits{i=\alpha}^\beta \dbinom{n}{i}=f(\...
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A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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55 views

Binomial Sum Formula

I can't find a good closed form expression for this, $\sum_{k=0}^n\left[\binom{n}{k}\binom{m}{k}\right]$, where n is the variable, and m is a fixed constant, to be included in the formula. :( Can ...
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68 views

Inequality with power function and binomial coefficients

Any suggestion on how to proceed to show: $$\frac{2(m+1)^m -1 }{(m+1)m} - \sum_{k=0}^{m} {{m}\choose{k}} \frac{m^k}{(k+1)^2} >0 $$ where $m\geq 2$ is of course an integer. Numerical results ...
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How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $

Let $a\in\mathbb N$, and $b\in\mathbb R, b\geq 1$ How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $
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A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot \binom{D+1}{i+...
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Binomial random variable/z-score question?

I was given the problem: In a restaurant called ”Allegory”, on average 1 in 10 people order a bottle of white wine. Out of a sample of 50 people 11 chose a bottle of white wine. Has this wine become ...
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Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

For a given k, ${n\choose k}$ is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding base p digit of k (consider the p-ary notation for n ...
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39 views

Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
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How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
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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 $\...
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how to represent a binomial coefficient in terms of a series?

I have to find the power series for (n+m C m) or (n+m C m ) - 1 i.e representing it in some sort of power . Is it even possible ? P.S. :- Thanks in advance .
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Truncated Binomial Series

Can the truncated binomial series be expressed as a closed form \begin{align} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{k} x^{k} \end{align}
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The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial :...
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How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the second ...
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Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n \...
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176 views

Sum concerning Catalan triangle and binomials

I am trying to prove the following relation. For $k \in \mathbb{N},k \geq 2$ and $i=1,\ldots,k-1$ \begin{equation} {2k-2-i \choose k-1-i}=\sum \limits_{l=0}^{k-1-i} 2^{2l} T(k-2-l,k-1-i-l)-2 \sum \...
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66 views

Maximizing expected value when distribution is binomial

Consider the following problem: $$\max_{n\in\mathbb N}\;f(n)= \frac12 \left[v_0 \sum_{i=\lceil k_n \rceil}^n \binom{n}{i}p^i (1-p)^{n-i} + v_1\sum_{i=1}^{\lfloor k_n \rfloor}\binom{n}{i}q^i(1-q)^{n-i}...
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74 views

General solution for a combinatorial problem

I want to find a general solution for a problem. I explain the problem with an example. $\underline{Problem}:$We have a matrix $A$ of size $M \times N$, where $M <N$. We choose sub-matrices of ...
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63 views

$\mathbb{E}(\alpha^Y)$, where $Y$ is negative binomial

It is given that $\alpha>0$ and that \begin{equation} \mathbb{P}(Y=y)=\begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^kp^y \end{equation} are there any ideas how to calculate expected value of $\...
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56 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...