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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Which one is greater $600!$ or $300^{600}$

Which one is greater $600!$ or $300^{600}$ $\bf{My\; Try::}$ I have used Stirling Approximation. For large $n>2\;,$ We can write $\displaystyle n! \approx \left(\frac{n}{e}\right)^n\sqrt{2\pi n}$...
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60 views

How to find the value of sum of even binomial coefficient? [duplicate]

I want to find the value of $\sum_{i=0}^k {n \choose 2i}$ where $2k\,\le\,n$. Is there any short formula to find the answer.
2
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1answer
24 views

How to expand an expression containing $x^2$ by binomial theorem

I understand that the coefficient of, say $x^8$, in the expansion of $(1+x)^{10}$, would be ${10 \choose 8}$, but what about an expression like $(1+x^2)^{10}$? Would I have to square root the ${10 \...
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1answer
75 views

How we can show this ;$\frac{x^2+y^2+z^2}{2}\times\frac{x^5+y^5+z^5}{5}=\frac{x^7+y^7+z^7}{7}$ [closed]

Let be $\quad x+y+z=0$ show this: $\frac{x^2+y^2+z^2}{2}\times\frac{x^5+y^5+z^5}{5}=\frac{x^7+y^7+z^7}{7}$ I solved ,but Im interesting what are you thinking about this,how can we arrive to ...
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56 views

How is $\lim_{x \to a}(\frac{x^n - a^n}{x - a}) = n\times a^{n-1}$?

In my book this is termed as a theorem and the proof given is as follows :- $\lim_{x \to a}(\frac{x^n - a^n}{x - a})$ = $\lim_{x \to a}(\frac{(x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^...
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5answers
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Counting arguments Given one prove the other identity

Given: $${n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n} = 2^n$$ Prove the following in 2 ways. $$ {n \choose 1} + 2 {n \choose 2} + 3 {n \choose 3} + \cdots + n{n\choose n} =...
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42 views

Counting arguments question about sums of binomial coefficients [duplicate]

Use counting arguments to prove these identities: I don't know how to type this: it is two numbers in brackets the first on top of the other, but there is no fraction line. Here is an image of both of ...
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2answers
51 views

Checking an identity involving binomial coefficients

I need some help to check the following identity: for every $0\leq i\leq l\leq r$ $$\sum_{j=0}^i\binom{r-l+i-j}{i-j}\binom{l-i+j}{j}=\binom{r+1}{i}.$$ Is this true ? Answering to John, this ...
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2answers
85 views

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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1answer
66 views

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
36 views

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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3answers
76 views

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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1answer
26 views

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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1answer
14 views

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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38 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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0answers
32 views

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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2answers
84 views

Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even [duplicate]

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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57 views

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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33 views

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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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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1answer
54 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
33 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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1answer
78 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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1answer
106 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 triangle

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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1answer
25 views

$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$

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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922 views

A strange combinatorial identity: $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [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
49 views

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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1answer
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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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1answer
37 views

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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31 views

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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80 views

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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1answer
29 views

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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2answers
53 views

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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33 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
123 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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1answer
34 views

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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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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3answers
52 views

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!