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 binomial coefficients multiplied

How to approximate the next one relative to $m$ $$ \sum_{k=0}^m \binom {n}{n-k} (n-k-\frac k{\sqrt{2}})^2? $$ Or for example the simplier sum $$ \sum_{k=0}^m \binom {n}{n-k} (n-k)^2? $$
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187 views

Unusual binomial sum: $\sum_{d=k}^{n} {d \choose k} p^{d}(1-p)^{n-d}$

Does anyone know how to simplify the following sum? It's been giving me and everyone else I've showed it to quite a bit of trouble. I'm quite confident that this should simplify, but I just can't seem ...
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72 views

Relation between Hyperfactorial, Superfactorial, Pascal's Triangle and Binomial Coefficient

I read here that the product of the elements in the $N^{th}$ row of Pascal's triangle is equal to $(n!)^{n+1}/(\prod_{k=1}^n k!)^2$. Let's call the product of elements in the $i^{th}$ row of Pascal's ...
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1answer
73 views

Rising factorial power

How the expression below can ve proved: $(a + b)^{\overline{n}} = \sum\limits_{j=0}^{n}C_n^j a^{\overline{n-j}}b^{\overline{j}}$ where $x^{\overline{n}}$ - is rising factorial power: ...
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1answer
156 views

How to solve the equation $\sum_{n=1}^\infty n^{-2}/\binom{n+x}{n} =\frac{3}{2}$ for $x$?

I find this problem on this page. Find $x\in\mathbb{R}$ such that $$\sum_{n=1}^\infty \frac{1}{n^2\displaystyle\binom{n+x}{n}}=\frac{3}{2}$$ Thank you very much.
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4answers
113 views

Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$

Prove the identity: $$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$ I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$ but got stuck along the way.
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1answer
38 views

Proof of identity involving binomial coefficient

My homework assignment was essentially to prove the following binomial coefficient identity: Prove $$\begin{pmatrix} -n \\ k\end{pmatrix} = (-1)^k \begin{pmatrix} n+k-1 \\ k\end{pmatrix}$$ ...
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2answers
45 views

Solve easy sums with Binomial Coefficient

How do we get to the following results: $$\sum_{i=0}^n 2^{-i} {n \choose i} = \left(\frac{3}{2}\right)^n$$ and $$\sum_{i=0}^n 2^{-3i} {n \choose i} = \left(\frac{9}{8}\right)^n.$$ I guess I could ...
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95 views

Tricky binomial coefficients problem

Let $k$ be a positive integer and let $n = 6k - 1$. Let $$S(n)=\sum_{j=1}^{2k-1} (-1)^{j+1} {{n}\choose{3j-1}}$$ How do you prove that $S(n)$ is never zero?
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Can one interchange the limit and summation in this example?

Let $L$ be a positive real number and $a$, $b$ and $x$ real numbers strictly between $0$ and $L$. For integers $m$ and $n$, define $$ A_{m,n} := \sum_{k=1}^{[\frac{\sqrt{n}(b-a)}{2}]} \frac{1}{2^n} ...
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2answers
33 views

Clarification of a proof of an identity for binomial coefficients, $\binom{n}{k}= \binom{n-1}{k-1}+\binom{n-1}{k}$

I am studying analysis 1 (first term). So there is this definition in our textbook: for every $n \in \mathbb{N} \ge 1$ and every k $\in \mathbb{Z}$ there is $\binom{n}{k}= ...
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1answer
29 views

Finding the Binomial Coffecient

Is there a way to find the binomial coefficient of $x^{14}$ in $$(x^0+x^1+x^2+x^3+x^4)^6$$ I tried to use sum of G.P form but it did'nt help.
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Nice formula for $\sum_{m=0}^n{2m\choose m}{2(n-m)\choose n-m}$? [duplicate]

I am trying to find a nice formula for \begin{align}\sum_{m=0}^n{2m\choose m}{2(n-m)\choose n-m}\tag{1}.\end{align} After failing to simplify it, I asked WolframAlpha (see link), and apparently, it ...
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139 views

Calculate $\lim_{n\rightarrow +\infty}\binom{2n} n$

Calculate $$\lim_{n\rightarrow +\infty}\binom{2n} n$$ without use Stirling's Formula. Any suggestions please?
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1answer
98 views

Tricky Negative Binomial example

Let $Y$ count the number of widgets succesfully produced before $r$th failure. We are told that machine shuts down when $30$th failure has occured, that is $r=30$. Then probability of producing $y$ ...
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1answer
51 views

$\mathbb{E}(\alpha^Y)$, where $Y$ is negative binomial

It is given that $\alpha>0$ and that \begin{equation} \mathbb{P}(Y=y)=\begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^kp^y \end{equation} are there any ideas how to calculate expected value of ...
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3answers
174 views

Binomial Problem : Find the coefficient of this equation!

How to find $x^{13}$ from this equation: $\left ( x^{3}+1 \right )^{2}\left ( x^{2}-\frac{2}{x} \right )^{8}$ I'm very confused with these equation and don't know how to solve using Binomial Newton ...
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1answer
55 views

Expressing sum using simple formula (without summation)

Express by a simple formula not containing a sum: $$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k}$$ I figured that $$\sum\limits^{n}_{k=1} \binom{k}{m}\frac{1}{k} = ...
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46 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
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1answer
115 views

summation of a finite sequence?

What is the summation of the finite sequence: $$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}} {2i - 2}\\ {i - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {2n + 2 - 2i}\\ ...
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27 views

A multinominal coefficient type identity

While trying to prove the power series expansion of $(1-z)^{-m}$ “by hand”, I'm stumped trying to prove the following identity $$\sum_{k=0}^m (-1)^k {{(m+n-k-1)!} \over {k!\, (n-k)! \,(m-k)!}}=0,$$ ...
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4answers
183 views

A closed form for sum of binomial coefficients

What is a closed form of the sum: $$\binom{n}{0}+\binom{n-1}{1}+\binom{n-2}{2}+\binom{n-3}{3}+\cdots$$ A combinatorial proof would also be much appreciated. Any general techniques to solve such sums ...
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1answer
41 views

Prove $\binom{N}{k}=\frac{N^k}{k!}\left(1+O(\frac{1}{N})\right)$ when $N \rightarrow\infty$

Can someone help prove the following: $\binom{N}{k}=\frac{N^k}{k!}\left(1+O(\frac{1}{N})\right)$ when $N \rightarrow\infty$. Thanks in advance!
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1answer
28 views

Maximize parial sum of binomials

During my research I got to the point when I need to find $$ \arg \max_w \left( (n-w) \sum_{j=0}^d \binom{w}{j} \binom{2^r - (j+1) 2^{r-j-1}-2}{t} \right) $$ with respect to $w$ only (i.e. $d$, $n$, ...
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1answer
67 views

function from a Binary sequence to the Natural Numbers

I apologize if this is a duplicate question. I don't know enough terminology to thoroughly search. However, given a sequence of binary numbers $1_10_20_3...0_n$, $0_11_20_3...0_n$, ... , ...
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37 views

difficulties in a proof with binomial theorem

how can one prove, that: $\sum_{k=0}^{n}\binom{2n+1}{k}=2^{2n} $ I was trying to use the binomial theorem, but I do have difficulties with 2n+1 in the binomial coefficient. Thank you
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41 views

Value of sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$

What is the value of the sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$? It looks like the binomial expansion $(1+x)^n=\sum_{i=0}^n\binom{n}{i}x^i$, but we only take every other term, and also the power is ...
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4answers
192 views

proving an invloved combinatorial identity

How to prove following Identity? $$\sum_{k=0}^n (-1)^k {n-k \choose k} m^k (m+1)^{n-2k} = \frac {m^{n+1}-1}{m-1}, m \ge 2$$ This seems very hard to me. Any idea about how to prove it combinatorialy? ...
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1answer
60 views

Is the following limit correct? $[\lim_{n\to\infty}\binom{n}{50}(\frac2n)^{50}(1-\frac2n)^{n-50}]$

$$\lim_{n\to\infty}\binom{n}{50}\left(\frac2n\right)^{50}\left(1-\frac2n\right)^{n-50}$$ Taking $nh=1$ and $K=\binom{n}{50}\left(\frac2n\right)^{50}\left(1-\frac2n\right)^{n-50}$, we have: ...
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$\sum_{k=0}^n\binom nk x^k=\sum_{k=0}^n\binom nk x^{n-k}.$

$$\sum_{k=0}^n\binom nk x^k=\sum_{k=0}^n\binom nk x^{n-k}.$$ I want a deeper understanding of solving problems in this nature. I can´t grasp this writing way.I get confused just by looking at these ...
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199 views

How to prove that $\sum_{k=0}^n\cos(k\pi)\binom{n}{k}=0$?

$$\sum_{k=0}^n\cos(k\pi)\binom{n}{k}=0$$ I approached this problem having no idea that $\cos(k\pi)$ could be substituted so easily. I tried first to expand the sum to $n$. So I wanted to ask if ...
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How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_{2},\cdots,k_{d}}^2$

Question: Assmue that $d$ is give postive integer numbers,and $$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots ...
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Can binomial coefficient be defined as a natural number if n is the cardinality of a countable set?

Can binomial coefficient ${n \choose k}$, $k$ less than or equal to $n$, be defined as a natural number if $n$ is the cardinality of a countable set?
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3answers
37 views

Summation using cdf of binomial distribution

I'm trying to find the exact value of the following equation: $$\sum_{x=0}^{1000}\tfrac{x^{2}-x+5}{x!(1000-x)!}2^{x}7^{1000-x}$$ I've managed to convert to the following: ...
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2answers
54 views

Relation among numbers in a triangle like tartaglia:

I have a polynomial associate to a number: $$\begin{align} k&=1 &n-1\\ k&=2 &n^2-2n +2\\ k&=3 &n^3-3n^2+6n+6\\ k&=4 &n^4-4n^3+12n^2-24n+24 \end{align}$$ and in ...
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Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ ...
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Sum of coefficients in binomial theory.

While trying to get introduced to binomial theory at university's website, I learned about the sum of binomial coefficients, and they showed me some of the features, and one of them was the pyramid of ...
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38 views

Chance of winning in a raffle

A raffle consists of 10 sheets with 10 numbers (1 to 10) on each sheet i.e. 100 chances in total. The draw is done by first selecting a sheet at random and then selecting the winning number out of the ...
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76 views

Proving that polynomials with rational coefficients have integer roots

Obviously, polynomials with integer coefficients will satisfy P(x)$\in$ Z or every x $\in$ Z. But how do we prove that those with rational coefficients can produce integer roots? For instance, I have ...
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Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
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1answer
99 views

Compute $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$

I want to calculate $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$. No idea in my mind. Any help? Context I want to calculate the expected value of bits per symbols in adaptive arithmetic coding when the ...
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1answer
110 views

Prove a theorem in combinatorics

I want to show that for $k=1,...,(n-1)$ we have : $\binom{n}{k}\leq \frac{n^n}{k^k(n-k)^{n-k}}$ I have used induction on $k$, but I have not deduced the above relation.
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63 views

Calculate limit involving binomial coefficient

How can I calculate this limit. Let $p\in[0,1]$ and $k\in\mathbb{N}$. $$\lim\limits_{n\to\infty}\binom{n}{k}p^k(1-p)^{n-k}.$$ Any idea how to do it ?
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2answers
93 views

Binomial transform

how can be prove expression: $\sum \limits_{s = 0}^{2k} (-1)^s\binom{n+s}{n}\binom{n+2k-s}{n} = \binom{n+k}{k}$ by using this identity: $(1 − t)^{−n−1}(1 + t)^{−n−1}= (1 − t^2)^{−n−1},$ or how ...
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61 views

Showing the equality of two rook polynomials.

I'm reading Barbeau's Polynomials. I've done the following: Taking an arbitrary chessboard $C$ with some of the squares forbidden (with $n$ being the number of squares and $F$ the number of ...
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4answers
81 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
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1answer
36 views

Proving the binomial coefficients by induction (half-done, but need help)

Defining the binomial coefficients $n \choose k$ as follows, i) for all $n \in \mathbb{N}$, $\binom{n}{0} = \binom{n}{ n} = 1$ (ii) for all $2 \leq n \in \mathbb{N}$ and for all $ 1 \leq k \leq n-1, ...
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0answers
60 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
3
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1answer
63 views

Proving a certain sum is 1 [duplicate]

I'd like to prove that, for any $M,N\in\mathbb{N}$ with $M\leq N$, and any $n\in\mathbb{N}$ with $n\leq M$, the sum: $$\sum\limits_{k=0}^n\frac{\binom{M}{k}\!\!\binom{N-M}{n-k}}{\binom{N}{n}}=1.$$ I ...
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3answers
81 views

Find $k$ given that ${14 \choose k} = {14 \choose k-4}$

Ok, so I stumbled upon the question on the title these days, when going over Apostol's Calculus I. Now, because of the placement of the question in the exercises section, I'm convinced that the book ...