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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185 views

Asymptotics of ${2^n \choose n}$?

How can one compute the asymptotics of ${2^n \choose n}$? I know it is bounded below and above by $\left(\frac{2^{n}}{n}\right)^n$ and $\left(\frac{2^{n}e}{n}\right)^n$. If I plug in Stirling's ...
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3answers
206 views

Coefficient of $x^7$ in binomial expansion of $(1/6-3x)^{17}$

How would I determine the coefficient of $x^7$ in the expansion of $(1/6-3x)^{17}$ and show the answer as a fraction?
3
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3answers
183 views

General Leibniz rule for triple products

I have a question regarding the General Leibniz rule which is the rule for the $n^{th}$ derivative of a product and reads: $$ (f g)^{(n)}=\sum_{k=0}^{n} {n \choose k} \,f^{(k)} g^{(n-k)}. $$ ...
2
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1answer
69 views

Proving Bernoulli's Inequality for $h<0$

I'm answering question 19 of chapter two of Spivak's Calculus and I can't seem to think of a way of doing it. I don't want to look up the answer so I thought I'd ask for a hint as to the general ...
0
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1answer
109 views

Binomial coefficients identity (sum of the powers of the natural numbers)

I've found exercise with binomial coefficients in Kostrikin's book. Proof that $\sum_{i=1}^n{{r+1}\choose{i}}\left(1^i+2^i+\dots+n^i\right)=(n+1)^{r+1}-(n+1)$ I was trying to check that for ...
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3answers
106 views

Understanding $n \left(\frac{2n \choose n}{4^n}\right)^2$ for large $n$

My answer over at cstheory.stackexchange.com involved the expression $$\lim_{n\to \infty} n \left(\frac{2n \choose n}{4^n}\right)^2$$ According to Wolfram Alpha, this expression is at most ...
2
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1answer
43 views

Solving the recurrence $x_n = \frac{\sum_{i=0}^{n-1} x_i}{2n}$

In my answer over at cstheory.stackexchange.com I set variables according to the recurrence $$x_0=1, x_n = \frac{\sum_{i=0}^{n-1} x_i}{2n}$$ Wolfram Alpha tells me that apparently the solution to this ...
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2answers
78 views

Reducing ${n\choose k} - {n\choose k-1}$

I'm writing computer program which on some point has to compute following formula: $${n\choose k} - {n\choose k-1}$$ Because I have following limits: $$n \le 4000, \space k \le\frac{n}{2}$$ computing ...
9
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2answers
196 views

An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the beginning of the respective chapter of Jeffrey Mishlove book “The Roots of ...
2
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6answers
107 views

$\sum_{k=1}^m k(k-1){m\choose k} = m(m-1) 2^{m-2}$

Thanks for those who helped me yesterday. I have another one and I'd appreciate if you can shed some light upon to keep me going. This one is related with binomial theorem. $$\sum_{k=1}^m ...
4
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1answer
87 views

Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
3
votes
3answers
216 views

Find the 100th derivative of $x \sinh(2x)$

If $f(x) = x \sinh(2x)$, find $f^{({100})}(x)$. My (Incorrect) working so far: Using Leibniz' Formula for derivatives: $$(fg)^{(n)}=\sum_{k=0}^n{n\choose k}f^{(k)}g^{(n-k)}$$ ...
5
votes
3answers
288 views

Given n $\in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
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1answer
150 views

Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots

Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...
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1answer
59 views

Prove that if $a_n = \sum_{r=0}^n \frac1{C_{r}^n}$, then $\sum_{r=0}^n\frac{r}{C_{r}^n} = \frac12na_n$

Prove that if $a_n = \sum_{r=0}^n \frac1{C_{r}^n}$, then $\sum_{r=0}^n\frac{r}{C_{r}^n} = \frac12na_n$ Tried to use nCr = $\frac{n}{r}$(n-1)C(r-1) But this doesn't seem to work.
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1answer
139 views

Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
6
votes
3answers
100 views

Show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$

In this question we are asked to show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$ What I did: $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \sum_{k=2012}^{n} 2^k*1^{n-k}\binom{n}{k} \leq ...
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4answers
66 views

Binominal Theorem

Could anyone help me with homework or give me a hint? Any help would be highly appreciated. Given a set of N distinct objects: How many ways are there to pick any number of them to be in a pile ...
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3answers
156 views

Probability of choosing coins from a bag (why doesn't binomial coefficient work?)

Studying for my exam and would appreciate some help. I have a bag with 2 pennies, 1 nickel and 1 dime. I pick 2 at random. The solutions say: Pr(PP) = $\frac{2}{4} \times \frac{1}{3}$ = ...
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2answers
51 views

A hotel bought some tvs, some are not working…

I am working through a stat book trying to freshen up on my math. Here is a problem that is posed... A shipment of 10 televisions includes three that are defective. In how many wats can a hotel ...
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2answers
100 views

Upper bound for $\sum_{j=0}^i {i \choose j}^{n}$

Is there an upper bound for sums of powers of binomial coefficients? I have $$\sum_{j=0}^i {i \choose j}^{n}$$ where $n$ is a positive integer. I am hoping this will help me solve Limit for a ...
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1answer
317 views

Divisibility of binomial coefficient by prime power - Kummer's theorem

Let's say we have binomial coefficient $\binom{n}{m}$. And we need to find the greatest power of prime $p$ that divides it. Usually Kummer's theorem is stated in terms of the number of carries you ...
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4answers
522 views

Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
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votes
1answer
23 views

Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
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1answer
62 views

Simple Random Walk

How to find: $\lim\limits_{N \to \infty}\sum\limits_{m=0}^Nu_m$ where $u_m$=${2m \choose m}p^mq^m$ I know there are two cases to consider depending if $p$ and $q$ are equal or not. I should probably ...
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2answers
73 views

generating function and binomial distribution - counting

I am trying to understand generating function. I have the following problem: There are 50 students in the International Mathematical Olympiad (IMO) training programme. 6 of them are to be selected to ...
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2answers
53 views

Upper bound for sum

I am trying to get an upper bound the following sum: $$S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r} .$$ Any hints would be greatly appreciated. I thought of using ...
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3answers
178 views

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

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

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
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2answers
40 views

Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
2
votes
2answers
222 views

Alternating sum of binomial coefficients $\sum(-1)^k{n\choose k}\frac{1}{k+1}$ [duplicate]

I would appreciate if somebody could help me with the following problem Q:Calculate the sum: $$ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} $$
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4answers
127 views

Find $\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$

Find $$\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$$ I got $$\frac{2^{2n+1}(2n^2+n+1)-1}{(2n+1)(2n+2)}$$ but the answer is $$\frac{2^{2n+1}(2n^2-n+1)-2}{(2n+1)(2n+2)}$$ Thanks for the help...
4
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0answers
124 views

Combinatorial Identity

I have to validate the following identity which is defined: $$ \sum_{k=1}^n (-1)^{k-1}*q^{\frac{k(k-1)}{2}} *\frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 $$ where $0<q<1$. I ...
2
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1answer
68 views

Combinatorics - finding coefficients when summing over permutations of permutations

I have $N$ 2-tuples. Each tuple* can either be up, in which case it has components $(a,b)$, or it can be down, in which case it has components $(c,d)$. Given that exactly $N_\mathrm{up}$ of these ...
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0answers
32 views

Analytical solution for binomial equation

Suppose that the random variable $X \sim \operatorname{Binomial}_{n,p}$, and suppose we have $p' \in [0,1]$. I have been asked to solve for the least $n$ such that $P(X \leq 2) = p'$. It was ...
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3answers
415 views

Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$

I'm looking for a reference with the proof of the following binomial identity: $$\sum_{k=0}^n \binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$$ I've looked in a number of textbooks that have a ...
2
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1answer
39 views

combinatorics - number of ways to choose r out of n (with inclusion-exclusion)

Quick question. Out of a set of $n$ apples, we are given that $m$ are delicious. Show that the number of different combinations to choose $r$ apples that contain all of the delicious ones is given by ...
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2answers
95 views

combinatorics - fixed point permutations

Simple question but I just need a little tip to finish it. we are given $A=\{1,2,3...,2n-1,2n\}$ the set of all integers between and including $1$ and $2n$. We are asked how many different ...
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3answers
590 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
3
votes
1answer
116 views

Pascal Triangle general formula

I'm working on a presentation on the Binomial Theorem for my Algebra 2 class and while writing Pascal's Triangle, I came across one of the properties that I haven't seen in a while. That being ...
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1answer
61 views

Two team playoff question, homework help

I am working through my college text book (mathmatical statistics freund/walpole) trying to refresh my stat skills. Its been a couple years... I would sure appreciate any pointers on an exercise ...
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1answer
76 views

Relation between Binomial coefficient and Stirling number of second type

Is that true, that for every n,k such that $$k>1$$ we have the inequality $${n \choose k} \leq {n \brace k}$$?
4
votes
3answers
121 views

How many numbers $k$ of $200 \choose k$ are divisible by $3$? $k \in \{0,1,2,\cdots 200\}$

"How many of the numbers (200 Choose k), where k is an element of the set {0,1,2,3,4,....,200} are divisible by 3? " Here is my thinking: (200 Choose 0,1, and 2) are not multiples of 3 but every ...
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1answer
61 views

Express number of ways integer can be written as coefficient in generating series

Question: "Express the number of ways that an integer $n$ can be written as a sum of a cube of an integer $s\ge-1$ plus the fourth power of an integer $t$ plus the square of an odd integer $r$ as a ...
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2answers
124 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
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0answers
125 views

Combinatorial word problems (Discrete math)

I have a problem with writing the word problems to which the answers are the following expressions. I am not sure if these answers sound right. I am not good with writing questions to these ...
3
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0answers
44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
2
votes
2answers
359 views

Closed form expression for unusual sum of binomial coefficients

How do I get a closed form expression for $\sum_{i=c}^{n} i\binom{i}{c}$? Note that the index ranges over the upper values of the binomial, not the lower. I know computer algebra systems can give me ...
5
votes
4answers
270 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
2
votes
1answer
84 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...