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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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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1answer
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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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2answers
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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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1answer
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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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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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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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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 $f_1(x) = (x - 2) ^ 2$ ...
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Which one is greater $600!$ or $300^{600}$

Which one is greater $600!$ or $300^{600}$ $\bf{My\; Try::}$ I have used Stirling Approximation. For large $n>2\;,$ We can write $\displaystyle n! \approx \left(\frac{n}{e}\right)^n\sqrt{2\pi n}$...
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How to find the value of sum of even binomial coefficient? [duplicate]

I want to find the value of $\sum_{i=0}^k {n \choose 2i}$ where $2k\,\le\,n$. Is there any short formula to find the answer.
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1answer
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How to expand an expression containing $x^2$ by binomial theorem

I understand that the coefficient of, say $x^8$, in the expansion of $(1+x)^{10}$, would be ${10 \choose 8}$, but what about an expression like $(1+x^2)^{10}$? Would I have to square root the ${10 \...
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How we can show this ;$\frac{x^2+y^2+z^2}{2}\times\frac{x^5+y^5+z^5}{5}=\frac{x^7+y^7+z^7}{7}$ [closed]

Let be $\quad x+y+z=0$ show this: $\frac{x^2+y^2+z^2}{2}\times\frac{x^5+y^5+z^5}{5}=\frac{x^7+y^7+z^7}{7}$ I solved ,but Im interesting what are you thinking about this,how can we arrive to ...
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3answers
56 views

How is $\lim_{x \to a}(\frac{x^n - a^n}{x - a}) = n\times a^{n-1}$?

In my book this is termed as a theorem and the proof given is as follows :- $\lim_{x \to a}(\frac{x^n - a^n}{x - a})$ = $\lim_{x \to a}(\frac{(x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^...
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5answers
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Counting arguments Given one prove the other identity

Given: $${n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n} = 2^n$$ Prove the following in 2 ways. $$ {n \choose 1} + 2 {n \choose 2} + 3 {n \choose 3} + \cdots + n{n\choose n} =...
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2answers
51 views

Checking an identity involving binomial coefficients

I need some help to check the following identity: for every $0\leq i\leq l\leq r$ $$\sum_{j=0}^i\binom{r-l+i-j}{i-j}\binom{l-i+j}{j}=\binom{r+1}{i}.$$ Is this true ? Answering to John, this ...
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2answers
85 views

Show that $\sum_{r=0}^{n}\binom{n}{r}\binom{m+r}{n}= \sum_{r=0}^{n}\binom{n}{r}\binom{m}{r}2^r$

Here $\binom{a}{b}$ is the number of ways in which $b$ objects can be chosen from a collection of $a$ distinct objects. Show that: $$\binom{n}{0}\binom{m}{n}+\binom{n}{1}\binom{m+1}{n}+\binom{n}{2}\...
3
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1answer
66 views

Is there a rigorous proof of this combinatorial identity?

Theorem: For any pair of positive integers $n$ and $k$, the number of $k$-tuples of positive integers whose sum is $n$ is equal to the number of $(k − 1)$-element subsets of a set with $n − 1$ ...
2
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1answer
36 views

Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...