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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Function $f: \mathbb{Z} \to \mathbb{Z}^n$ related to $\sum_{k=1}^{x} k^n$.

The sequence $\{a_0,a_1,...a_x\}$ has closed form $a_n=\sum_{i=0}^{\infty} \Delta^i(0) {n \choose i}$ where $\Delta a_n$ denotes the operation mapping $a_n$ to $a_{n+1}-a_n$ and $\Delta^i(0)$ is ...
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Finding maximum value of ${n \choose r}$ for given value of n [duplicate]

While I was solving some binomial theorem chapter questions I encountered many questions which asked me me to find maximum value of ${n \choose r}$ for given value of n. Example: Find n for which $...
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Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form

I need to express $$1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$$ in a simplified form. So I used the identity $$(1+x)^n=1 + \binom{n}{1}x + \...
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Binomial identity in a finite field

Suppose we have a prime $p$ and consider $\mathbb{F}_q$ where $q=p^s$ for some $s$. Fix a positive integer $m \geq 2$ and let $t \leq m-1$. Let $r$ be a positive integer such that $0 \leq r \leq q^t-1$...
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Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

I'm trying to prove that \begin{equation} \sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1} \end{equation} with the Binomial Theorem. I know that the B.T. states that \begin{equation} (x + y)^n = ...
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Pattern with the the tetration of summations.

While dealing with a question with finding an explicit form for a sequence I noticed something: $$\sum_{x_0=0}^{n-1} 1=\frac{n}{1!}$$ $$\sum_{x_0=0}^{n-1} \sum_{x_1=0}^{x_0-1} 1=\frac{n(n-1)}{2!}$$ ...
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Is this quantity divisible by $p$?

Let $p$ be a prime. Let $$x_{1} = \binom{2p-1}{p}-1$$ $$x_{2}=\binom{3p-1}{2p}-\binom{2p-1}{p}$$ $$x_{3} = \binom{4p-1}{3p}-\binom{3p-1}{2p}$$ and I observed that for small values of $p$ $x_{1}$, $x_{...
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Trinomial Pascal's Triangle

I know that there's a trinomial theorem (and a multinomial theorem), but I was wondering if there was a similar structure for trinomials as there is for binomials, like Pascal's triangle. Thanks in ...
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Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$

Is there a closed form for the following? $$\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$$ how about a pretty limit for large $b$. I have tried using the binomial expansion for the $(b+ck)^...
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Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$ I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find ...
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What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

Since both $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor$ and $\sum_{i=0}^n \binom{n}{i}$ have simple closed-form evaluations, it is natural to consider the evaluation of the binomial sum $\sum_{...
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Binomial coefficient paths?

Here's a problem and my attempt to answer it: We want to get a binomial coefficient identity depending on grid walking. Starting from the bottom left corner and going to the top right corner. You can ...
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63 views

binomial coefficients difference? [closed]

I need a difference of 2 binomial coefficients that would be equivalent to the following sum: $12\choose5$+$11\choose5$+$10\choose5$+$9\choose5$+$8\choose5$ How to answer this?
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Decomposition of $ \binom {n} {j-1}j^k $

It is easy to check that: $$ \binom {n} {j-1}j = \binom {n-1} {j-1}+\binom {n-1} {j-2}(n+1) $$ and $$ \binom {n} {j-1}j^2 = \binom {n-2} {j-1}+\binom {n-2} {j-2}(3n+2)+\binom {n-2} {j-3}(n+1)^2 $$ We ...
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Coefficient of $x^{103}$ in the following multinomial expansion.

What is the coefficient of $x^{103}$ in the expansion of $$(1+x+x^2+x^3+x^4)^{199}(x-1)^{201}$$ ?. The answer is an integer between $0-9$. So I wrote the given expression as $(x^5-1)^{199}(x-1)^{2}$. ...
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Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
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binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
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Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
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Evaluate an increasing sum of binomial coefficients: $\sum_{k=1}^nk\binom{m+k}{m+1}$

I've been working on a problem and got to a point where I need the closed form of $$\sum_{k=1}^nk\binom{m+k}{m+1}.$$ I wasn't making any headway so I figured I would see what Wolfram Alpha ...
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Limit analysis for binomial series raised to 0 < fractional exponent < 1

I am trying to expand a series and trying to find a limit analysis: (1+x)^a where 0 < a < 1. I understand that a possible expansion is: 1 + ax + a(a-1)(x^2)/(2!) + a(a-1)(a-2)(x^3)/(3!) +... ...
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What's the formula to map between multiindices and indices?

What is the formula to map between multiindices and indices? By multiindex, I mean a variable $I\in\mathbb{N}^d$ where $|I|=\sum\limits_{i=1}^d I_i=n$. Here, $d$ denotes the dimension. Basically, ...
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Is there an identity related to $\binom{n-j-1}{k}+\binom{k+j}{k}\pmod{n}$?

I noticed that when $n$ is an odd prime, the following congruence $$\binom{n-j-1}{k}+\binom{k+j}{k} \equiv 0 \pmod{n}$$ holds for $0 \le j \le \frac{(n-k)}2$ and odd values of $k$ such that $0 < k ...
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For all positive integer $n$ prove the equality: $\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}^2}{k+1}=\frac{\binom{2n}{n}}{2n}$

For all positive integer $n$ prove the equality: $$\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}^2}{k+1}=\frac{\binom{2n}{n}}{2n}$$ My work so far: $$\frac{n\binom{n-1}{k}}{k+1}=\frac{n(n-1)!}{(k+1)k!(n-k-...
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If the last term of $(2^{1/3}-\frac{1}{\sqrt{2}})^n$ is $(\frac{1}{3^{5/3}})^{\log(\frac{8}{3})}$, what is the $5\rm{th}$ term from the beginning? [closed]

If the last term of $$\left(2^{1/3}-\frac{1}{\sqrt{2}}\right)^n$$ is $$\left(\frac{1}{3^{5/3}}\right)^{\log(\frac{8}{3})}$$ then the value of $5th$ term from beginning is ?. So I simplified $(243)^{1/...
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Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
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Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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# of bit strings of length n (even>2), with n/2-1 zeros and n/2+1 ones, zero followed by one

case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one Answer: 3 case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, ...
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Binomial theorem incomplete expansion

I got this question and I am little bit confused, whether the question is correct or not. $n\choose0$-$ n\choose1 $+$ n\choose2$-$ n\choose3$+.........+$(-1)^r$$ n\choose r$=$28$ Now we are ...
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How to fast compute the coefficient of $x^{2^{n-1}}$ in polynomial $[\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]]^{2^{n-r+1}}$?

How to fast compute the coefficient of $x^{2^{n-1}}$ in polynomial $$\left(\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]\right)^{2^{n-r+1}}$$ or compute the coefficient of any term $x^k$ for $k\geq 0$? ...
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Formula for $\sum_{i\geq 0} i{n \choose 2i}$?

So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone ...
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Summation of Binomial Coefficient: $\sum\binom{n+k}{2k} \binom{2k}k \frac{(-1)^l}{k+1}$

I am trying to solve this summation problem . $$\sum\limits_{k = 0}^\infty {\left( {\begin{array}{*{20}{l}} {n + k}\\ {2k} \end{array}} \right)} \left( {\begin{array}{*{20}{l}} {2k}\\ k \end{array}} \...
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Binomial Coefficient Identity Involving Summation

Prove that $$\sum_{j=0}^n (-1)^j \binom{n+j-1}{j}\binom{N+n}{n-j} = \binom{N}{n} $$ I tried to prove this via binomial expansions of $(1-x)^N (1+x)^{-m}$, and equating the coefficients of $x$, ...
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A sum of squared binomial coefficients

I've been wondering how to work out the compact form of the following. $$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}$$
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Multiple objects, Number of combinations

A bag contains colored balls: 8 red balls 4 white balls 4 blue balls 4 green balls 2 purple balls 2 orange balls 1 yellow ball 1 black ball Total of 26 balls. I'd like to determine the number of ...
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What is the probability that when flipping a fair coin ten heads will be thrown in a row

I just cant figure this out.. Should I be using binomial distribution? Chance of getting ten tail long series in 100 coin throws
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Triangular numbers and pascal's trangle

The following are the triangular numbers. rank = 1 2 3 4 5 6 term = 1 3 6 10 15 21 A rule for triangular numbers is: ...
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Finding the coefficient of $x^7$ in the expansion $(1 + x)^{23}$

By definition, the Binomial Theorem states: $$(x+y)^n = {n\choose 0}x^n + {n\choose 1}x^{n-1}y + {n\choose 2}x^{n-2}y^2 + \cdots + {n \choose {n-1}}xy^{n-1} + {n \choose n}y^n$$ For any $x,y\in\...
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Research: Looking for a sequence that produce variation's of Pascal's triangle

Prologue I am an undergraduate so if my terminology or approach seem inappropriate/confusing please explain in the comments. I created a notation where $$F(0 \rightarrow n,x) = [\hspace{1mm}F(0 ,...
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Why are the coefficients equal in expansions for $(1+x)^{m+n}$ and $(1+x)^m (1+x)^n$?

I don't understand a step of a solution: Let $m,n\in\mathbb{N}$ and $r\in\{1,\dots,m+n\}$ then $$(1+x)^{n+m}=\left(\sum\limits_{i=0}^m \binom{m}{i}x^i\right)\left(\sum\limits_{j=0}^n \binom{n}{j}x^j\...
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On the proof of Lucas' theorem

Lucas theorem states that Let $m,n$ be two natural numbers, $p$ be a prime. Suppose that $m, n$ admit the following base $p$ representation $$m=m_0+m_1p+\cdots+m_sp^s,\qquad n=n_0+n_1p+\cdots+n_sp^...
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How to express $\binom{a+b}{n}$ as a sum of regular coefficients

I am trying to prove that $(1 + f(x))^a(1 + f(x))^b = (1 + f(x))^{a+b}$ in the world of formal power series. At a certain point in the prove I get \begin{align*} (1 + f(x))^a(1+f(x))^b& = \...
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A question on divisibility of binomial coefficient

In this paper, page 3, theorem 4, the author claimed that If $m, n, k$ are three positive integer such that $\text{gcd}(n, k)=1$ then $\binom{mn}{k}\equiv 0\pmod n$. And he proved it as ...
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Intuitive ways to get formula of binomial-like sum

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
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28 views

Calculating Probability Of Value With Dice [duplicate]

I am trying to write a program to calculate the probability of a number of dice thrown equaling a specif value. I have done some working out in excel, to try and find a patter, but I am at a loss. At ...
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The coefficient of $x^{100}$ in the expansion of $(1-x)^{-3}$.

Please solve it and tell me the technique so that I can solve it in examination in multiple choice questions.
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Estimation of suitable significance level for high numbers of experiments

I am a scientist and work with high numbers of experiments. For example: I have tested $ 1000 $ different parameters between $2$ groups. I found $10 $ parameters, which are significantly different ...
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Binomial coefficients mod p

I want to find the following sum mod $p$ (prime number): Let $i\geq \frac{p-1}{2}$, $ \sum_{k=i}^{p-1} \binom{k}{i}\binom{k}{p-i-1} \pmod{p} $ OK, I succeeded in simplyfying this argument to the ...
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104 views

Sum of combinatorics sequence $\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1}$

I need to find sum like $$\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1},\qquad \text{ for even } n$$ Example: Find the sum of $$\binom{20}{1} + \binom{20}{3} +\cdots+ \binom{20}{19}=\ ?$$
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1answer
59 views

Orthogonal combinatoric sum

I have verified this identity in Matlab: $$ \sum_{k=m}^n~(-)^{n+k}\frac{2k+1}{n+k+1}\binom{n}{k}\binom{n+k}{k}^{-1}\binom{k}{m}\binom{k+m}{m}=\delta_{nm} $$ Where $n, m$ are positive integers. It was ...
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1answer
37 views

Solving a Binomial Expansion Question

I have a question which asks to find the coefficient of x and the constant term, for $f_n(x)$ given that $f_1(x) = (x - 2) ^ 2$ ...