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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Sum of products of binomial coefficient $-1/2 \choose x$

I am having trouble with showing that $$\sum_{m=0}^n (-1)^n {-1/2 \choose m} {-1/2 \choose n-m}=1$$ I know that this relation can be shown by comparing the coefficients of $x^2$ in the power series ...
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3answers
177 views

Integral with binomial coefficient

Is it possible to evaluate this integral without using the gamma function $$ \int_0^1 {a \choose b}x^b(1-x)^{a-b} dx$$ It looks a little like part of binomial theorem, but I don't have an idea how to ...
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1answer
84 views

An inverse binomial summation.

I am looking for a closed form for this summation: $$ \sum_{j=1}^m\frac{r^{-j}}{j{m\choose j}} = \sum_{j=1}^m\frac{r^{-j}}{m{m-1\choose j-1}} = \frac1{rm} \sum_{k=0}^{m-1}\frac{r^{-k}}{{m-1\choose k}} ...
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Asking About Binomial Sum Related to Fibonacci

How would I prove $$ \sum\limits_{i,j\ge 0} {n-i \choose j} {n-j \choose i}=F_{2n+1} $$ where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of Fibonacci numbers? Thank you very ...
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1answer
41 views

How do you calculate a binomial distribution with k > R as opposed to k = R

I'm given the formula: $\displaystyle P(X = k; n, p) = \binom {n}{k} * p^k * q^{n-k}$ And we need to work out the binomial coefficient by hand, instead of using C(n,r). So I have a question: "Some ...
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56 views

How find this sum $\sum_{k=0}^{p}t^k\binom{n}{k}\binom{m}{p-k}$

Find the closed form $$\sum_{k=0}^{p}t^k\binom{n}{k}\binom{m}{p-k}$$ since $$\binom{n}{k}\binom{m}{p-k}=\dfrac{n!}{(n-k)!k!}\cdot\dfrac{m!}{(p-k)!(m-p+k)!}$$ then I can't
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0answers
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Proving $\sum_{k=1}^{n}\binom{n-1}{k-1}{\binom{n+k}{k}}^{-1}=\frac 12$ combinatorially

Question : How can we prove the following equations combinatorially? $$\begin{eqnarray}\sum_{k=1}^{n}\frac{\binom{n-1}{k-1}}{\binom{n+k}{k}}&=&\frac ...
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2answers
67 views

Evaluate the sum $\sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k}$

Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ...
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2answers
72 views

Combinatorial proof of $n! = {n\choose k}k!(n-k)!$

Can someone give me some insight on the proof of $$n! = {n\choose k}k!(n-k)!$$ I understand algebraically why they are equal but I'm having trouble seeing what the right side is actually saying. On ...
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Why does this sum of binomial coefficient ratios equal 1?

In the course of doing some calculations comparing unrepeatable sets of event trials, I ended up with the following identity. If my reasoning and my math are correct then this ought to be true, and ...
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1answer
57 views

Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$ [duplicate]

I'm trying to prove this combinatorially. $$k\binom{n}{k} = n\binom{n-1}{k-1}$$ I know the first step is to relate a question to the equation. My question was if you have $n$ friends how many ways can ...
4
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5answers
192 views

Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$

I've seen the identity $\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ used here recently. I checked for proofs here ...
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1answer
25 views

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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0answers
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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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1answer
42 views

Evaluating Combination Sums

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 playing numbers, I conjectured ...
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3answers
115 views

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
35 views

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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2answers
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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
18 views

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
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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
69 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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0answers
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Binomial expansion, the greatest term…

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
189 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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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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2answers
44 views

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
87 views

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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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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1answer
92 views

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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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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29 views

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
66 views

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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0answers
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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
55 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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68 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
92 views

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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2answers
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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
44 views

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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1answer
44 views

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
30 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 ...
4
votes
4answers
108 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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0answers
33 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
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1answer
25 views

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 ...
2
votes
3answers
167 views

Sum of binomial coefficients with three variables

What's the sum of coefficients of $(a+b+c)^8$? Thanks in advance!