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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Show that $\sum_{r=0}^{n}\binom{n}{r}\binom{m+r}{n}= \sum_{r=0}^{n}\binom{n}{r}\binom{m}{r}2^r$

Here $\binom{a}{b}$ is the number of ways in which $b$ objects can be chosen from a collection of $a$ distinct objects. Show that: $$\binom{n}{0}\binom{m}{n}+\binom{n}{1}\binom{m+1}{n}+\binom{n}{2}\...
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Is there a rigorous proof of this combinatorial identity?

Theorem: For any pair of positive integers $n$ and $k$, the number of $k$-tuples of positive integers whose sum is $n$ is equal to the number of $(k − 1)$-element subsets of a set with $n − 1$ ...
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1answer
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Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...
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Proof that combinations are equal to coefficients in the binomial expansion

Let $n\in N$, $k\in Z$, $o\leq k \leq n$. Define $C^{n}_k$ as the coefficient of $x^{n-k}y^k$ in the expansion of $(x+y)^n$ $$(x+y)^n= \sum^{n}_{k=0} C^{n}_k x^{n-k}y^k$$ Prove that ${C^{n}_{k}}={...
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Finding domain of binomial function under square roots

What are the conditions that must be taken care of when finding the domain of the function $\sqrt {\binom {x^2+4x}{2x^2+3} }$ ?
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Help to prove an expression about sums of binomials coefficients using Complex Power Series theorem.

I'm solving some exercises from Kreszig's Advanced Math book and I got stuck in one: (10th ed, chapter 15.3, problem 18): Using $(1+z)^p*(1+z)^q=(1+z)^{p+q}$, obtain the basic relation: $$\sum_{n=0}^...
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Combinatorial formulas and interpretations

I found that $$ \sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
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23 views

How to prove that ${l \choose a_1,…,a_n}\le n^{l-1} $ , when $a_1+…+a_n=l$.

In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then $${l \...
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Proof of Pascal's rule by induction [duplicate]

I'm trying to prove pascal's rule by induction.Who could tell me how can I prove the following equation. i.e $$\dbinom{n}{k}=\dbinom{n-1}{k-1}+\dbinom{n-1}{k}$$ Many Thanks!!!
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Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even

If $n$ is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
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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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Binomial Theorem coefficient sum…

Recently I encountered a question but its answer as well as the way the author of the book has solved the question seemed wrong to me.. Find the sum of the coefficients of the expansion of $$...
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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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1answer
44 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
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1answer
32 views

identity on Pascal's triangle modulo 2

Consider Pascal's triangle with entries modulo $2$, and let $(k,l)$ denote the $l$-th entry in the $k$-th row by $(k,l)$. Show that, for all $n \in \mathbb{N}$, each entry of the triangle with ...
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77 views

Prove that $\sum_{j=0}^{n}H_j{n\choose j}^2={2n\choose n}\left(2H_n-H_{2n}\right)$

Let $H_n$ the $n$th Harmonic numbers and $H_0=0.$ Prove that $$\sum_{j=0}^{n}H_j{n\choose j}^2={2n\choose n}\left(2H_n-H_{2n}\right)$$ I encounter this problem since 2012 and have verify ...
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Best way to expand $(2+x-x^2)^6$

I've completed part $(a)$ and gotten: $64+192y+240y^2+160y^3+...$ Using intuition I substituted $x-x^2$ for $y$ and started listing the values for : $y, y^2 $ and $y^3,$ in terms of $x$. $y=(x-x^2)...
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Find $k$ if given the constant term of a binomial expression?

Consider the expansion of $x^2(3x^2+\frac{k}{x})^8$. The constant term is $16,128$. Find $k$. This is simply an example of a type of question I cannot understand how to do. I have many questions: 1) ...
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100 views

continued fraction $F(x)$ that is a generating function of central binomial coefficients

Given the following continued fraction $$F(x) =\cfrac{1}{x+\cfrac{2^2(2^2-1)}{6x+\cfrac{3^2(3^2-1)}{12x+\cfrac{4^2(4^2-1)}{20x+\cfrac{5^2(5^2-1)}{30x+\ddots}}}}}=\frac{1}{\sqrt{x^2+4}}$$ Then $$\...
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Anti diagonal elements of table forming pascal traingle

A function in $k$ and $n$ leads to the formation of this table. The elements in this table are rows of pascal triangle if we look at the anti diagonals elements of this table. They have also been ...
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$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$

2$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=2\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$. This is an identity in a note for a class in Markov Processes, but I can't ...
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A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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2answers
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Problem based on sum of binomial coefficients

Let $m$ be the smallest positive integer such that Coefficients of $x^2$ in the expansion $\displaystyle (1+x)^2+(1+x)^3+.....+(1+x)^{49}+(1+mx)^{50}$ is $\displaystyle (3n+1)\binom{51}{3}$ ...
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Combinatorial proof of a certain alternating sum of binomial coefficients

The following identity appeared as a question earlier today $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 \end{...
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whether is $\sum_{k=0}^{j-1} \binom{i}{k}=\sum_{k=0}^{j-2} \binom{i-1}{k}+\sum_{k=0}^{j-1} \binom{i-1}{k}$ true or false?

I have tested some trivial samples when $j = 1,2,3$. But I can't prove if it is true or false generally. Any help would be great, thanks!
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Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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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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Seating children in the cinema

I just had finished my class and have been struggling with a problem. There's $9$ seats in the cinema, and two families $F_a=\{F_1,F_2,F_3,F_4,F_5\},$ $F_b=\{F_a,F_b,F_c,F_d\}$ In how many ways can ...
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Odd binomial sum equality has only trivial solution?

Suppose $$\sum_{k\ {\rm odd}}^n {n \choose k} 2^{(k-1)/2} = \sum_{k\ {\rm odd}}^m {m \choose k} 2^{(k-1)/2} 3^{(m-k)/2}.$$ Does $m=n=1$? Clearly $m \leq n$, and for every $n$ there is at most one $m$....
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Please help me compute this$ \sum_m\binom{n}{m}\sum_k\frac{\binom{a+bk}{m}\binom{k-n-1}{k}}{a+bk+1}$

Compute following: $$ \sum_m\binom{n}{m}\sum_k\frac{\binom{a+bk}{m}\binom{k-n-1}{k}}{a+bk+1} $$ Only consider real numbers a, b such that the denominators are never 0. Now I simplify it into $$ -\...
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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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T0 show an equation by using binomial theorem

$$\left(1+\frac{a}{n}\right)^{(n-k)} = e^a \left(1-\frac{a(a+k)}{2n}\right)+o\left(\frac{1}{n}\right)$$ as $n\to\infty$. How the binomial theorem show this above equality? Thank you for your help!
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Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$

I need to proof this following equality : $$\binom{m+n}{m}=\sum_{i=0}^m \left(\binom{m}{i}\binom{n}{i}\right)$$ This is what I did combinatoric proof: Left : subset with $m$ members from $m+n$ ...
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Coefficient of product of polynomials.

Suppose we have the polynomial $f(x)$ and another polynomial $g(x)$. How can I find the coefficient of say $x^n$ in the product of the polynomials without actually multiplying. I am not that ...
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Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
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How to choose $n$ balls from the bags?

Given $4$ bags A, B, C and D. Bag A contains 'a' number of balls. Bag B contains 'b' number of balls. Bag C contains 'c' number of balls. Bag D contains 'd' number of balls. I have another bag E ...
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Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
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30 views

Newton Binomial Problems

Let there be this binomial: $$ (\sqrt{2} + \sqrt[3]{3})^{8}$$ How many rational terms are there in it's development? I tought that the number of terms is given by n + 1 = 8 + 1 = 9, but that doesn't ...
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1answer
116 views

Can anyone verify $\int_{0}^{\infty}\frac{e^{-2nx}+2nx-1}{x(e^x+1)}dx=\ln{2n\choose n}$? [closed]

Central binomial coefficient from mathworld $$\frac{2^{2n+1}}{\pi}\int_{0}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx={2n\choose n}$$ Here we have $\ln{2n\choose n}$ in term of another integral, $$\int_{0}...
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Coefficient of $x^n$ in binomial expansion

I want to find the coefficient of $x^n$ in $G(x)$ where $ G(x) = \frac{1}{1-x^{a_1}}\times\frac{1}{1-x^{a_2}}\times\dots\times\frac{1}{1-x^{a_k}}$ how do I approach this? It would be helpful if it ...
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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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3answers
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Intuitive explanation of $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$

Could anyone please explain me the reasoning behind this formula? $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$ Thanks so much!
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What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$.

What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$. Here is what I got so far (using Cauchy's integral formula) : $$\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n} ~=~ \...
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1answer
43 views

Proving binomial identities [duplicate]

Can someone help me prove these two binomial identities using either walks in Pascal's triangle or a committee-selection model? $(1)$ $\qquad$ $\displaystyle\sum_{k=0}^m {m\choose k}{n\choose r+k}={m+...
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Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I assume....
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54 views

Probability of winning a prize in a raffle (that each person can only win once)

There is a raffle coming up. 4000 tickets have been sold, and there are 10 prizes to win. I have bought 8 tickets. What are the odds I will win a prize? Note: each person can only win once. There ...
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45 views

Binomial coefficient of power n

How can I find the coefficient of $x^n$ using binomial theorem? $$\frac{1-x}{(1+x)^3}$$
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Self-avoiding walks from one diagonal to the other on $mxn$ lattice is ${m+n \choose m,n} $

According to wikipedia "self-avoiding walks from one end of a diagonal to the other, with only moves in the positive direction, there are exactly $$ \binom{n+m}{n,m} $$paths for an $m × n$ ...
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1answer
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How can I find the cubic polynomial, using $4\times4$ linear system with coefficients? Or any help of reference?

part A in standard form $$y = a_0 + a_1t + a_2t^2 + a_3t^3$$ passes points $$(0, 4), (1, 3), (−1, 7), (2, −2)$$ part B and for the same cubic polynomial in shifted basis $$\{1, t − 2,(t − 2)^2,(...
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1answer
54 views

How many $5$ card poker hands contain at least $1$ red and $1$ black card?

How many $5$ card poker hands contain at least $1$ red and $1$ black card? I used inclusion-exclusion to calculate my answer. The number of total poker card hands are:$$52\choose 5$$I have $26$ red ...