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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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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43 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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1answer
31 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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38 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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Find the largest term for the binomial expansion of (n+1/n)^(2n+1)?

For (∀n ∈ N{1,2 }). So what I have done at first to try the (Tk+1/Tk)>=1, but it gave me wrong answers, for example I got that my coefficient was negative which is absurd.
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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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53 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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116 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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0answers
19 views

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
25 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
47 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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87 views

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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3answers
35 views

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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1answer
22 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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1answer
53 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
90 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
103 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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52 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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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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53 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
36 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
27 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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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 ...
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1answer
57 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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76 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 ...
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1answer
17 views

Binomial Coefficient as Sum of a Sum

Few days ago, I found this equation: $ \sum_{i=1}^n \sum_{j>i} \frac{1}{2} = {n \choose 2} \frac{1}{2} $ I didn't manage to prove it. Does anyone of you know how to prove it?
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An asymptotic expression of sum of powers of binomial coefficients.

Let $k$ be a fixed positive number and $n$ an integer increasing to infinity. Then $$\sum_{\nu =0}^n \binom{n}{\nu}^k \sim \frac{2^{kn}}{\sqrt{k}} \left( \frac{2}{\pi n} \right)^{\frac{k-1}{2}}.$$ ...
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56 views

The smallest $n$ for which the sum of binomial coefficients exceeds $31$

I have a problem with the binomial theorem. What is the result of solving this inequality: $$ \binom{n}{1} + \binom{n}{2} + \binom{n}{3} + \cdots +\binom{n}{n} > 31 $$
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Proof that $\sum\limits_{k = 0}^n {n\choose k} =2^n$ using Binomial Expansion Formula

HW problem here. Not sure how to even start on it. Prove that $$\sum\limits_{k = 0}^n {n\choose k} =2^n$$ Any help is appreciated. For Search purposes: (Hint: Use the binomial expansion mentioned ...
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135 views

How many Binary numbers?

How many binary numbers of length $n$ can be generated where $n > 7$ and the number either start with $000$ or end with $111$? My questions is, can I choose an $n$ randomly? For example, let's say ...
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137 views

Closed form of $\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$

Is there a closed form for: $$\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$$ where: $$h(x)=(1-x)^{\alpha}(A-Bx)^{\frac{1}{\gamma}-\alpha}$$ and ...
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80 views

combinatorial or algebraic proof of combinatorial identity

I would like to find out how to prove the following identity, assuming it is correct: $\displaystyle\sum_{r=0}^n\binom{n}{r}\binom{m+r}{l}=\sum_{r=0}^n\binom{n}{r}\binom{m}{r+l-n}2^r$ for ...
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1answer
27 views

Probability problem (Bernoulli trial)

I recently became interested in studying probability and I stumbled upon this question: There are three points: A, B and C. Exactly two paths exist between A and B and exactly two paths exist between ...
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53 views

Explanation about an identity involving inverse binomial coefficients.

Now, I was solving a this problem. It asks for summation of $$\sum\limits_{k =0}^\infty\dfrac{1}{{n+k \choose n}}$$ I solved it using this answer, the answer turns out to be $$\dfrac{n}{n-1}$$ ...
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75 views

Find summation of following series.

What will be the formula for following infinite series? $$1 + \frac{1!}{x+1} + \frac{2!}{(x+1)(x+2)}+ \cdots$$ $$ x\ge2 $$ up to infinite What pattern i got : coefficient of $ \frac{1!}{x+1}$ ...
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52 views

Proving a Binomial Identity

Can you please help me with problem 25. I need to prove that $f(n+1)=2 f(n)$, where $f(n)$ is the LHS of the expression, from there on I can do it my self. I have tried using the binominal theorem ...
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Find $f(n)$ in $\binom {2^n} {n^4} = (f(n)+ o(1))^n$

Task is to find $f(n)$ in the following equation: $\binom {2^n} {n^4} = (f(n)+ o(1))^n$ I've found that the problem is a bit over my head. I'm attaching my partial solution below: With use of the ...
3
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1answer
38 views

Growth of ratio of binomials polynomial or exponential?

Is the growth of $$ \dfrac{\binom{2n}{\sqrt{n}}}{\binom{n}{\sqrt{n}}} $$ polynomial or exponential (or other kind of growth) in $n$? I tried using the Stirling's approximation, which gives ...
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1answer
41 views

Evaluation of a limit of ratio of sums [closed]

How do I calculate the value of $$ \lim_{n\to \infty} \left(\frac{\sum_{r=0}^{n} \binom{2n}{2r}3^r}{\sum_{r=0}^{n-1} \binom{2n}{2r+1}3^r}\right)$$
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Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...
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76 views

Proof of an identity involving binomial coefficients

I have found numerically that the following identity holds: \begin{equation} \sum_{n=0}^{\frac{t-x}{2}} n 2^{t-2n-x}\frac{\binom{t}{n+x}\binom{t-n-x}{t-2n-x}}{\binom{2t}{t+x}} = ...
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4answers
272 views

Alternating sum of binomial coefficients multiplied by (1/k+1)

I'm trying to prove that $$\sum_{k=0}^n {n \choose k} (-1)^k \frac{1}{k+1} = \frac{1}{n+1}$$ So far I've tried induction (which doesn't really work at all), using well known facts such as ...
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60 views

How to calculate $\sum\limits_{k=0}^{n}{k\dbinom{n}{k}}$ [duplicate]

I derived this sum from a problem I have been working on. Somehow I don't know how to proceed. I only know some basics like $\sum\limits_{k=0}^{n}\dbinom{n}{k} = 2^n$. Meanwhile I am reading the ...
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1answer
30 views

Product of binomial coefficients

Is there any way to simplify given expression ($j$ and $i$ are given, $n\leq \lfloor j/i \rfloor$) $$\prod_{x=1}^n \binom {j-(x-1)i} {i}$$ (e.g. in terms of factorials)? Thanks!
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1answer
43 views

Black bears and tan-colored bears catching salmon in Alaska

One of popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all "black" bears are black, though- some are tan-colored. Suppose that 6 black bears ...
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0answers
61 views

Permutation and combination identity

Prove that $\displaystyle \sum_{i=0}^n \binom{n}{i}\binom{m+i}{n}=\sum_{i=0}^n \binom{n}{i}\binom{m}{i} 2^i$ for natural numbers $m,n.$ The question doesn't seem to have any direct combinatorial ...
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27 views

How can I solve for $n$ in this binomial coefficient equation

How can I solve for $n$ in this binomial coefficient equation? $${n\choose 3} = {n\choose 9}$$ When I try to expand it using factorials, I get a very, very long equation, involving $n-s$ up to $n^6$ ...