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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Properties of cumulative binomial distribution

Let $F\left(k, n, p\right) = \sum_{i=1}^k\binom{n}{i}p^i\left(1-p\right)^{n-i}$ denote the cumulative binomial distribution function. If $F\left(k, n, p\right)-F\left(k, n, p'\right) \geq ...
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Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
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3answers
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Evaluating Combination Sum $\sum{n+k\choose 2k} 2^{n-k}$

Evaluate $$\sum_{k=0}^n{n+k\choose 2k} 2^{n-k}$$ So im not really sure how to begin with this. I would imagine we start with dividing out $2^{n}$, but not really sure much past that
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A conjecture including binomial coefficients

Question: $$\sum_{k=1}^{n}k\binom{2n}{n+k}=\frac n2\binom{2n}{n}$$ is true for every $n\in \mathbb N$? If this is true, then how can we prove this? When I was with playing numbers, I conjectured ...
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Binomial expansion question. easy!

I'm trying to do the binomial expansion of $(x-2)^{1/2}$. How do you do it? As far as I'm aware the expansion only works for $(1+x)^n$. How could I get it in that form? Thanks.
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1answer
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Help with this hard recurrence relation question

Please help with this. Suppose $\{a_n\}$ satisfies $$a_n=(n+1)a_{n-1}-(n-2)a_{n-2}-(n-5)a_{n-3}+(n-3)a_{n-4},$$ and $a_0=a_1=1,a_2=a_3=0$. Please sort out the general form of $a_n$. I guess $a_n$ ...
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Question about Binomial Distribution

The chance of a rose flower blooming is .28. You are going to plant 5 rose flowers, what are the chances of 4 of them blooming? I was thinking the answer would be 35% since 28%x5=140 and 140/4=35. ...
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1answer
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Generalization of Binomial Coefficients Congruence

It is well known and not hard to prove that $\binom{pA}{pB}\equiv\binom{A}{B}\mod p$ where $p$ is a prime. Now, how can we extend to show that this congruence holds $\mod p^2$. Finally, can we extend ...
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2answers
52 views

A binomial random number generating algorithm that works when $ n \times p $ is very small

I need to generate binomial random numbers: For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads ...
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1answer
73 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
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Binomial expansion, the greatest term… [duplicate]

My question is related to binomial expansion, and more precisely the greatest term in expansion. Is it right that the formula for finding the greatest term is $$T_k\ge T_{k+1}$$? Now going to the ...
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1answer
208 views

Binomial Summation

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
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2answers
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Proving this binomial identity

I'm required to prove the following binomial identity: $$\sum\limits_{k=0}^l {n \choose k} {m \choose l-k} = {n+m \choose l}$$ I tried various arrangements but reached nowhere. Finally I turned to ...
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Let $C=\dfrac{1}{k}\left [\binom{2k-1}{k}-1 \right ]$ where $k \ge 3$. Show that $C\ge 3$.

I have a problem: Let $C=\dfrac{1}{k}\left [\binom{2k-1}{k}-1 \right ]$ where $k \ge 3$. Show that $C\ge 3$. Any help will be appreciated! Thanks!
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Prove that $^nC_0+^{n+1}C_1+^{n+2}C_2+…+^{n+k}C_k = ^{n+k+1}C_{n+1}$ for $k\ge1$ [closed]

Prove that $^nC_0+^{n+1}C_1+^{n+2}C_2+.....+^{n+k}C_k = ^{n+k+1}C_{n+1}$ for $k\ge1$
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1answer
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How to compute the sum $\sum_{r=0}^n \frac{(-1)^r}{{n \choose r}}$

Consider the sum $$\sum_{r=0}^n \frac{(-1)^r}{{n \choose r}}.$$ I know the sum is zero when $n$ is odd (pretty simple). The sum is $2-\frac{2}{2 + n}$ when $n$ is even. Can somebody provide a proof ...
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how to compute the variance of the random variable Y in binomial distribution?

sorry to bother but I just saw some slides provided by the harvard university. One of those show the binomial distribution with the VAR(Y)=$\frac{\pi(1-\pi)}{N}$ Im bit confused because usually I see ...
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0answers
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Approximation of the following mathematical formula

I have the following mathematical expression which I need to simplify: $$\mu^2\sum_{x=0}^{n}\left(\frac{\theta}{\mu}\right)^x\frac{1}{H_x}{n+a\choose x}$$ $\mu$, $\theta$, $D$, and $a$ are ...
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4answers
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Binomial coefficient equivalence

Can someone explain to me why these 2 formulas are equivalent? (n \choose k) = (n \choose n-k)
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Help with a Binomial Identity: $\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = 2^{n+1}$

The following is a problem from the 5th edition of Niven's An Introduction to the Theory of Numbers: Problem 23 of Section 1.4 asks us to prove that $$\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = ...
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1answer
39 views

Binomial coefficients(Concrete mathematics 5.39)

I have no idea what I could do with this problem. I tried to substitute, but failed.. Hope someone can help
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How find the sum $2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\frac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$

Find the sum $$2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\dfrac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$$ where $B_{i}$ is Bernoulli numbers. my idea: since ...
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1answer
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1Prove that limit n tends to infinity $1 + 2 \sum_{k=0}^n1/\binom{n}{k} = e^2$

Prove that limit n tends to infinity $1 + 2 \sum_{k=0}^n1/\binom{n}{k} = e^2$ I have not been able to proceed ..tried many things like ratio of nck and nc(k+1)...also opened it.!! Not able to ...
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the number of $m$-divisible subsets of an $n$-set

Let $\omega$ be a primitive $m^{th}$ root of unity. How can we use the binomial expansion of $(1+\omega)^n$ to find out the number of $m$-divisible subsets of an $n$-set. Actually, I mean, to find a ...
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1answer
62 views

Show $\large\sum\limits_{j=0}^{r}\binom{j+k-1}{k-1}=\binom{r+k}{k}$

Show $\large\sum\limits_{j=0}^{r}\binom{j+k-1}{k-1}=\binom{r+k}{k}$ Hint: Place $r$ balls in $m$ urns, in how many of this arrangements can you find $b$ balls in the first urn. I'm sure that ...
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2answers
101 views

Sum of product of binomial coefficents

So, I try to manipulate some series and this sum came up in the coefficients $$\sum_{k=m}^{l-n+m}\binom{k}{m}\binom{l-k}{n-m}$$ whith $l\ge n$. I've seen the identities of the binomial coefficients ...
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1answer
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How prove this inequality $\frac{1}{n}\sum_{k=0}^{n-1}\binom{k}{a}\binom{k}{b}\le\frac{1}{a+b+1}\binom{n}{a}\binom{n}{b}$

let $a,b,n$ be positive integer numbers,and such $a,b\le n$, show that $$\dfrac{1}{n}\sum_{k=0}^{n-1}\binom{k}{a}\binom{k}{b}\le\dfrac{1}{a+b+1}\binom{n}{a}\binom{n}{b}$$ this inequality maybe ...
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5answers
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Formula for $\sum_{k=0}^n k^d {n \choose 2k}$

If $d \geq 1$ is an integer, is there a general formula for $$\sum_{k=0}^n k^d {n \choose 2k}\,?$$ We know that $\sum_{k=0}^n k {n \choose 2k} = \frac{n2^n}{8}$ and $\sum_{k=0}^n k^2 {n \choose 2k} = ...
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1answer
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Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
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Is $n\binom{\epsilon n}{t}>t\binom{n}{t}$ for large $n$ and fixed $\epsilon$ and $t$

Let $\epsilon$ and $t$ be fixed numbers with $t$ and integer. I came across the following inequality in a counting problem. $$n\binom{\epsilon n}{t}>t\binom{n}{t}.$$ I want to show that for $n$ ...
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1answer
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How prove $\binom{n}{m}\le\left(\frac{en}{m}\right)^m$ [duplicate]

Show that $$\binom{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ where $0<m\le n,m,n\in N^{+}$ My idea: since ...
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6answers
988 views

How do I begin proving this binomial coefficient identity: ${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$

This is a homework question. I'm asked to prove the identity: $${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$$ (The sum ends with ${n\choose n} = 1$, with the sign of the ...
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1answer
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Addition of Binomial Coefficients

$$\left[\binom n{k-1} + \binom nk\right] + \left[\binom nk + \binom n{k+1}\right] = \binom{n+1}k + \binom{n+1}{k+1}$$ Can anyone else explain to me, without using Pascal's triangle, how this ...
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1answer
54 views

Binomial coefficients in series

Here's a tricky one which I don't know how to start so any help would be appreciated. Show that no 4 consecutive binomial coefficients can be in AP and no 3 consecutive binomial coefficients can be in ...
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4answers
133 views

Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

An exercise in the first chapter of Discrete Mathematics, Elementary and Beyond asks for a proof of the following identity: $$ {n \choose 2} + {n+1 \choose 2} = n^2 $$ The algebraic solution is ...
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1answer
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Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
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Sum of binomial coefficients with three variables

What's the sum of coefficients of $(a+b+c)^8$? Thanks in advance!
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number of terms in the expansion containing powers of $x$

How do i find the number of terms containing powers of $x$ in the expansion of: $$(1+x)^{100}(1+x^2-x)^{101}$$ I tried using $(1+x)((1+x(1+(x)^2-x))^{100})$ which simplified into : ...
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Is there a simpler way to compute this sum?

For any given positive integers $m$, $n$ and $q$, such that $m\leq n$ the following sum $$S_p= \sum_{p=0}^m(-1)^{p+q} \binom mp \binom mq \binom np \binom nq\frac{p! q! (m+n-p-q)!}{m! n!}$$ is equal ...
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2answers
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Integral equation involving Binomial distribution

I am trying to find the form of a function $u^{(n)}(p)$ which satisfies $\forall k \in [0,n] \int_0^1 dp\, u^{(n)}(p) \binom{n}{k} p^k(1-p)^{n-k} = \frac{1}{n+1}$. This is a private case of a more ...
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1answer
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Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
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limit of $2n$-th root of $2n$-th central binomial coefficient — $\lim_{n\to \infty}{2n \choose n}^{\frac{1}{2n}}$

$\lim_{n\to \infty}{2n \choose n}^{\frac{1}{2n}} = 2$ according to wolframalpha. Does anyone see how to get this limit of $2$? Is it a numerical estimate or analytically exact?
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1answer
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Proof on Divisibility of Binomial Coefficients

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$ n \nmid {n \choose i} $$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.
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Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
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Inequality with two binomial coefficients

I am having trouble seeing why $$ \binom{k}{2} + \binom{n - k}{2} \le \binom{1}{2} + \binom{n - 1}{2} = \binom{n - 1}{2} $$
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3answers
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Set of solutions for a binomial inequality

I bumped into the following inequality: $${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$ Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using ...
2
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1answer
27 views

Finding the value of an expression involving co-efficients in binomial expansion

Let $$(1+x)^n = C_0 +C_1.x+C_2.x^2 +: : :+C_n.x^n,$$ n being a positive integer. Then find the value of the following expression: $$(1+C_0/C_1)*(1+C_1/C_2)*.....*(1+C_{n-1}/C_n)$$
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2answers
41 views

Show that if $a,k\in \mathbb{Z}$ with $0\leq k \leq a$, then $\binom ak=\frac{a!}{k!(a-k)!}=\binom {a}{a-k}$.

I'm reading Ghorpade's A Course in Calculus and Analysis. Given $a\in \mathbb{R}$ and $k\in \mathbb{Z}$, the binomial coefficient associated with $a$ and $k$ is defined by: $$\binom ak = ...
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6answers
286 views

Prove that $\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$

Prove that: $$\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$$ i know that: $$\sum_{k=0}^n {n \choose k} = {2^n}$$ how to get the (n + n^2)?
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1answer
55 views

Very loose bound on sum of first binomials

Let $n\geq k\geq 2$. Is it always true that $$\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\leq n^k?$$ The left-hand side is dominated by the term $\dfrac{n^k}{k!}$, so the statement should be true. ...