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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85 views

The probability that after repeated random drawing from an urn, all balls left in the urn will be red

Problem An urn contains $p$ red and $q$ green balls. Balls are drawn one by one till balls left in the urn are all red. Prove that the probability of this event is $\dfrac {p}{p+q}$. Please note that ...
11
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4answers
2k views

Combinatorial proof of a binomial coefficient summation.

Let $n$ and $k$ be integers with $1 \leq k \leq n$. Show that: $$\sum_{k=1}^n {n \choose k}{n \choose k-1} = \frac12{2n+2 \choose n+1} - {2n \choose n}$$ I was told this is supposed to use a ...
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3answers
326 views

Combinatorial Proof Of Binomial Double Counting

Let $a$, $b$, $c$ and $n$ be non-negative integers. By counting the number of committees consisting of $n$ sentient beings that can be chosen from a pool of $a$ kittens, $b$ crocodiles and $c$ emus ...
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1answer
107 views

Understanding a proof of the fact that $\binom{n}{k}$ is always a natural number.

Original source of question and solution. Question is on the left, answer is on the right. Question: Notice that all the numbers in Pascal's triangle are natural numbers. Use part (a) to prove by ...
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1answer
44 views
+50

Binomial Congruence Modulo a Prime

Let $p$ be a prime and $a, b$ natural numbers such that $1 \leq b \leq a$. I am trying to prove that $$\binom{ap}{bp} \equiv \binom{a}{b} \pmod p.$$ Furthermore, I have been tasked with proving that a ...
9
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4answers
345 views

Evaluate a finite sum with four factorials

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$ \sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
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0answers
30 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
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1answer
54 views

An upper bound and simplification for expression

I would like to find the upper bound (or simplification) of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}{n+1\choose j}{n \choose i}/{2n+1 \choose j+i}$$ where ...
3
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1answer
32 views

How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?

The sum is $$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}} $$ The first thing I thought of was the binomial coefficient. So I re-indexed it ...
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0answers
23 views

Proving an inequality having binomial coefficients

Suppose that $0 \leq b \leq b+x < a$. How could I prove the inequality \begin{equation} \left(\frac{a-b-x}{a-x}\right)^x \leq \cfrac{\binom{a-x}{b}}{\binom{a}{b}} \leq \left(\frac{a-b}{a}\right)^x ...
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0answers
16 views

Lower bound for a relative of the central binomial coeff

The central binomial coefficients $\binom{2m}{m}$ have g.f. $\frac{1}{\sqrt{1-4z}}$ and lower bound $\frac{4^m}{\sqrt{4m}} \le \binom{2m}{m}$. I'm interested in a related integer series $$T(2m, m) = ...
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3answers
84 views

Simplifying $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}$

In trying to simplify my answer to a problem posted recently, I am trying to show that $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}=8\binom{24}{4}$. I know that ...
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0answers
33 views

A sum with binomial coefficients in the numerator and denominator.

I am struggling with a combinatorial sum as apart of a long statistical-mechanics derivation. I would appreciate any help. I seek the result of the following summation, for integer $\ell,n$, and ...
2
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1answer
41 views

Simplifying a generating function in two variables with two binomial coefficients

I'm trying to to make the below expression simpler, and it would be great if it could be expressed as something like $(x+y)^k$. $$ \sum_{i=0}^k\binom{n+1}i\binom{m+1}{k-i}x^iy^{k-i} $$ The number ...
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2answers
41 views

Upper bound on $ \binom{a}{m+1}\sum ^m_{j=0} \binom{a-m-1}{j}/\binom{b}{j+m+1}$

Given $a,b,m$ such that $0<2m<a<b$. I would like to find out upper bound of $$S = \binom{a}{m+1}\sum ^m_{j=0} \frac{\binom{a-m-1}{j}}{\binom{b}{j+m+1}}$$ Anyone can help me please? Thank you ...
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7answers
149 views

Calculating $\displaystyle\sum_{i=1}^{n} \binom{i}{2}$

Show $\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and ...
0
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1answer
21 views

Probability $\sum_{j=n+1}^{2n+1} {M \choose m+1}{M-m-1 \choose j-m-1}/{N \choose j} $

I have a prob. problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and $n$ ...
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1answer
154 views

How prove this $\sum_{k+j=n,0\le k,j\le n}\binom{2k}{k}\binom{2j}{j}=4^n$ [duplicate]

Show that $$ \sum_{k\ +\ j\ =\ n\atop{\vphantom{\LARGE A}0\ \le\ k,\phantom{A} j\ \le\ n}}{2k \choose k}{2j \choose j} = 4^{n} $$ I think use integral solve it. But I don't it,and this problem is ...
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3answers
134 views

Compute $\sum_{k=0}^{1006}\binom{2013}{2k}$ using mathematical induction.

$$ \mbox{Compute}\quad\sum_{k=0}^{1006}{2013 \choose 2k} $$ Hi, I know that I have to use induction but I am kinda stuck here. any tips or suggestions would be great! Thanks
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3answers
304 views

Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$

Is it possible to write this in closed form: $$\sum_{k=0}^{n} k\binom{n}{k}\log\left(\vphantom{\Huge A}\binom{n}{k}\right)$$ Can you get something like $$n2^{n-1}\log(2^{n-1})$$
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2answers
52 views

closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$

how to get closed form for $$\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$$ I tried to write binominal in term of gamma function but I got no result what is your suggest to solve the problem ?
12
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3answers
356 views

How prove this $\sum_{k=0}^{n} \frac{\binom{2n-k}{n}}{2^{2n-k}}=1 $

Show that $$\sum_{k=0}^{n}\dfrac{\binom{2n-k}{n}}{2^{2n-k}}=1$$ I think this problem can be solved with nice methods, such as algebraic ones. Or can I use probability methods? Thank you
1
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2answers
142 views

How find this sum $\sum_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$

Find the sum close form $$f(x)=\sum_{i=0}^{2n}\dfrac{\binom{2n}{2i}\binom{2i}{i}x^{2i}}{2^{2i}}$$ if we let $$\dfrac{x}{2}=y$$ then $$f(y)=\sum_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$$ ...
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3answers
123 views

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
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3answers
36 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
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2answers
27 views

Prove Maximum term in the expansion.

How should I prove the maximum term in the expansion of $(x+a)^n$ where $ax>0$ is the term $C(n,r)x^{(n-r)}a^r$ for which $r= \left[\cfrac{(n+1)}{(n/a)+1} \right]$ ?
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1answer
58 views

Simplifying the sum of two consecutive binomial coefficients

I've been trying to figure out how they simplified the right side of the equation all the way down. $$\eqalign{ \binom nk + \binom n{k-1} & = \dfrac{n!}{(k)!(n-k)!}+\dfrac{n!}{(k-1)!(n-k+1)!} \\ ...
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2answers
60 views

Simplify the expression of binom

Any one knows how to simplify this expression or finding upper bound of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}$$ where $0<a<1$ is constant. Thanks a lot.
3
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1answer
67 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
5
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3answers
591 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
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1answer
30 views

Upper bound of $\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}$

I would like to find max (or sup.) of the sum: $$S=\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}.$$ I found $S\le \frac{1}{\sqrt{\pi n}}.2(n+1).4^n$ but It seems it's ...
1
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1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
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1answer
262 views

Manipulation of Geometric Series and Binomial Theorem

I was just hoping to confirm that the following manipulations make sense: Say I begin with $\frac{1}{(1-x)^n}$. Then we have $(1-x)^{-n} = $$\sum$ $-n\choose k$ $(-x)^k$ = $\sum$ $(-1)^k$ $n+k-1 ...
4
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2answers
64 views

Ordered partitions of an integer (with a twist)

I would like to know how to prove (preferably algebraically) that $P_1(2,n)=F_{2n+1}$, where $P_1(2,n)$ is what I define to be the number of ordered partitions of an integer, where the number $1$ has ...
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2answers
38 views

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 : ...
3
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2answers
265 views

This question involves Pascal's Triangle & Binomial Theorem. Full Question

Could anyone please help me on the following problem: Factorize the expression $P(n)=n^x-n$ for $x=2,3,4,5$ Determine if the expression is always divisible by the corresponding $x$. If divisible use ...
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2answers
16 views

Finding a particular term of a binomial expansion

How do I find the term involving $x^{10}$ in the expansion of: $$ (3+2x^2)^7 $$ I know from the binomial theorem that: $$ u_{n+1} = {^nC_r a^{n-r}x^r} $$ and that $n=7, a=3, x=2x^2, r=10$ in this ...
3
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2answers
40 views

Binomial dependent on a Poisson

I have been working on a problem with a binomial rv dependent on a poisson rv and have worked through to this point: $P(X=x) = \sum_{n=x}^{\infty} \dfrac{n!}{x!(n-x)!} p^x(1−p)^{n−x} ...
3
votes
1answer
38 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
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1answer
111 views

Combinations mod $n$ property

So after some "fooling around" I came across this property in Pascal's triangle (which seems to repeat, and makes a lot of sense): $\begin{pmatrix} n \\ k \end{pmatrix} \mod n = \begin{cases} n ...
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0answers
21 views

Probability in the urn model without replacement.

In an urn with $p$ total marbles, $p_A$ are white and $p-p_A$ are black, we know that the probability of drawing at least $m_A$ white marbles out of a $m$ without replacement follows the cumulative ...
15
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3answers
368 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
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1answer
24 views

Finding $V(X)$ when you don't have a density/distribution function.

I just did the first part of this problem: You have a lot of $50$ items and are taking a sample size of $15$. In the lot $3$ items are defective. The lot is accepted if the number of defective items, ...
2
votes
1answer
65 views

simplifying a triple sum of products of binomial coefficients

Right now I have a horribly-looking triple sum ($x,y,z$ are non-negative integers and $x+y+z=N$): $$ W_{12}(x,y)=\frac{x}{N}\sum_{l=0}^{x-1}\sum_{l'=0}^{y}\sum_{l''=0}^{z}{x-1 \choose ...
3
votes
1answer
48 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
3
votes
1answer
125 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
1
vote
1answer
134 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
1
vote
5answers
129 views

$\sum_{k=0}^{n}\frac{1}{(k+1)(k+2)}\binom{n}{k}=?$

I was asked to find a closed formula for the sum $$\sum_{k=0}^{n}\frac{1}{(k+1)(k+2)}\binom{n}{k}$$ could anyone give me an advice on how to get started?
7
votes
4answers
379 views

Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$

The other day a friend of mine showed me this sum: $\sum_{k=0}^n\binom{3n}{3k}$. To find the explicit formula I plugged it into mathematica and got $\frac{8^n+2(-1)^n}{3}$. I am curious as to how one ...
1
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3answers
173 views

Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$

I need hint proving this $$\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$$