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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Binomial identity $\sum_{i=0}^{n-1-j}\binom{n}{i+j}\binom{i+j}{j}(-1)^i=\binom{n}{j}(-1)^{n+j+1}$

Let $n$ be a positive integer and fix a non-negative integer $j\le n-1$. Is it true that $$ \sum_{i=j}^{n-1}\binom{n}{i}\binom{i}{j}(-1)^i=\binom{n}{j}(-1)^{n-1} $$ or, equivalently, $$ \sum_{i=0}^{n-...
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Expressing Factorials with Binomial Coefficients

Expression I have somehow stumbled upon this expression (I believe I have proved it, but that is not important right now), which I have tried to simplify by writing it like something like this (I ...
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Binomial coefficients of $p^a-1$ mod p

I need to show that $$\binom{p^a-1}{k}\equiv (-1)^k\mod p$$ where $p$ is a prime, $a$ is a positive integer, and $k<p^a-1$. I think I have done most of the proof, but I'm stuck at the very end. ...
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1answer
78 views

Where does this equation for $n\binom{\binom{n-1}{2}}{m}$ come from?

I'm reading a proof and am having trouble seeing why the following two lines are true: \begin{align*} n\frac{\binom{\binom{n-1}{2}}{m} }{ \binom{\binom{n}{2}}{m}} &= n \left(\frac{n-2}{n}\right)^...
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summation of binomial coefficients with squares

What is $$50^2\frac{{n\choose 50}}{{n\choose 49}}+49^2\frac{{n\choose 49}}{{n\choose 48}}...1^2\frac{{n\choose 1}}{{n\choose 0}}$$. i.e. $$\sum_{k=1}^{50} \frac{k^2\binom n k}{\binom n {k-1}}= ?$$ $...
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46 views

What is $x$ in a polynomial?

I know this sounds like an easy question. But I've never been told this. Suppose we have a polynomial... $$3x^2 + 5x - 9$$ I know $3$, $5$ and $-9$ are coefficients. I also know $2$ (form $3x^2$), ...
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Finite sum with three binomial coefficients

I need to find a closed form expression, if there is one, of the following sum: $$\sum_{j=0}^m{n+1-k\choose j}{k-1\choose m-j}{A+2-k+m-j\choose m-j+2}$$ where all parameters are integers, $~1\leq k\...
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6answers
217 views

Simplify $\binom nr+ 4\binom n{r+1} + 6\binom n{r+2} + 4\binom n{r+3}+\binom n{r+4}$ [closed]

For $$4\le r \le n,$$ $${n \choose r}+4{n \choose {r+1}}+6{n \choose {r+2}}+4{n \choose {r+3}}+{n \choose {r+4}}$$ equals 1. $${n+4}\choose{r+4}$$ 2.$${n+4}\choose{r}$$ 3.$${n+3}\choose{r-1}$$ 4.$${n+...
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Binomial expansion of $(x_{1}+x_{2}+…+x_{k})^{n} $ [duplicate]

If we expand $$ (x_{1}+x_{2}+...........+x_{k})^{n} $$ How many terms will be there once we collect terms with equal monomials? What is the sum of all coefficients? I literally have no clue how to ...
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53 views

how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients?

If you expand $(x_1+x_2+\cdots+x_k)^n$, how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients? I'm kind of lost here. This came up with ...
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1answer
63 views

Looking for a nonrecursive formula for the general derivatives of the quotient of functions

I want to prove that the $k$-th derivative $h^{(k)}(x)$ of the function $h(x)=\frac{1}{1+x^2}$ is zero at $x=0$ for all integer values $k>0$. My only idea was to go the stubborn way applying ...
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1answer
35 views

Find $\sum r\binom{n-r}{2}$

Let $A=\{1,2,3,\cdots,n\}$. If $a_i$ is the minimum element of set $A_i$ where $A_i\subset A$ such that $n(A_i)=3$, find the sum of all $a_i$ for all possible $A_i$ Number of subsets with least ...
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67 views

How do you deal with fractions in a binomial?

If I have something like this $$\binom{\frac{x}{k}}{\frac{y}{k}}$$ (where there are two fractions in a binomial but they have the same denominator) can I simplify this at all?
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42 views

What does this binomial sum equal?

I'm trying to evaluate this sum: $$\sum_{k=0}^n {n \choose k}{{2n+1}\choose k}$$ I thought I could work with generating functions of the two binomials. I know $$\sum_k\binom{n}k{}x^k=(1+x)^n$$ is the ...
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How to prove this equality about Eulerian numbers?

I want to prove the following equality where $A(k,m)$ is the Eulerian number : $$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$ I previously proved ...
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0answers
41 views

A theorem about binomial coefficient module prime

For any integer $r$ and prime $p$, there is a integer $n$ which $\binom{2n}{n}\equiv r \pmod{p}$. I tried Lucas's theorem, but I was stuck. Suppose $r\neq 0$, otherwise we can let $n=p$. Let $n=\...
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42 views

Ho can I find the sum of this sequence?

There is a sequence $$ F_n= \begin{cases} aF_{n-1}+q^{n-2}F_{n-2},& n \text{ is even}\\ bF_{n-1}+q^{n-2}F_{n-2},& n \text{ is odd} \end{cases} $$ with the initial conditions $F_0 = 0 ...
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42 views

Prove that if $0 \le k \le \frac {n-1}{2}$, then ${n \choose k} \le {n \choose k+1}$, with equality holding if and only if $k = \frac{n - 1}{2}$

Prove that if $0 \le k \le \frac {n-1}{2}$, then ${n \choose k} \le {n \choose k+1}$. Further, prove that equality is met if and only if $k = \frac {n-1}{2}$ I tried to use the contrapositive $${n \...
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1answer
41 views

Combinatorial proof (catching fishes and eating some from them)

Combinatorial proof for : $$ \binom{n}{m} \cdot \binom{n-m}{k-m} = \binom{n}{k} \cdot \binom{k}{m} $$ where $m\leq k\leq n$ I tried but I have no idea about how to do it. Any help will be ...
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64 views

Sum of the coefficients of polynomial $f(x)= (3x-2)^{107} (x+1)^4$

Sum of the coefficients of polynomial $f(x)= (3x-2)^{107} (x+1)^4$ Please hint me with this. I can't manage anything except taking 3 common from first bracket.
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1answer
21 views

Anti-symmetric ways

A car dealer lines up his best objects for sale. He has 'n' Porsche and 'n' Ferrari. How many anti-symmetric ways are there to arrange these cars? (Anti-symmetric means that if ith from left is a ...
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1answer
85 views

Find the sum of the coefficients in the expansion of the given expression using Binomial Theorem. [closed]

The expression is : $$(1-3x+x^2)^{111}$$ I tried treating $$(3x+x^2)$$ as one term to turn it into a binomial expression and expanding it to a few terms to see if i could find some pattern to use ...
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4answers
120 views

How to calculate $\sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i}$

how can we calculate this?$$ \sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i} $$ It is alternating and contains the Binomial coefficients which are given in terms of factorials as, $$ \...
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How to derive this binomial identity?

I believe the following is an identity (I've tested with a few random $m$ and $n$ values, could be wrong though): $$\sum_{k= 0}^{\infty}{m \choose k}{n \choose k}k=n\binom{m+n-1}{m-1}$$ but I'm not ...
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141 views

What is this binomial sum?

I'm trying to figure out what this sum is equal to: $$\sum^n_{k=0}k \binom{m-k}{m-n}$$ I thought there are n turns and on each turn you pick 1 object from k objects ($\binom{k}{1}=k$) and also pick $...
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122 views

How to prove the following binomial identity

How to prove that $$\sum_{i=0}^n \binom{2i}{i} \left(\frac{1}{2}\right)^{2i} = (2n+1) \binom{2n}{n} \left(\frac{1}{2}\right)^{2n} $$
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How do I find all the coefficients of $x^0$ in this: $(x-\frac{2}{\sqrt x})^8$

How do I find all the coefficients of $x^0$ in this: $(x-\frac{2}{\sqrt x})^8$ I got to $8 - k + (-0.5)k = 0$ and then $ 16 - 3k = 0 $ and thus, I can't find $k$ that will solve this equation. What ...
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187 views

upper bound formula the binomial coefficients with real valued arguments

I'm trying to prove the following. Let $n\in\mathbb{N},m\in\mathbb{N}\cup\{0\},\alpha\in (n-1,n)$ and $N\in\mathbb{N}:N\ge m+1$. Prove that \begin{align} &\sum_{k=N+1}^\infty\Big{|}\binom{n+m-\...
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162 views

Trying to solve the equation $\sum_{i=0}^{t}(-1)^i\binom{m}{i}\binom{n-m}{t-i}=0 $ for non-negative integers $m,n,t$

While considering a previous unanswered question, I started looking for the non-negative integer solutions $ m,n,t , (n\ge m)$ to the equation: $$ S(m,n,t)=\sum_{i=0}^{t}(-1)^i\binom{m}{i}\binom{...
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36 views

Sum of Sequentially Spaced Binomial Terms

Understanding that if $k>n$, we have that $\binom{n}{k}=0$, has there been any success coming up with closed formulas or asymptotic formulas for the following... $$B(n,k,j)=\sum_{i=0}^n\binom{n}{...
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114 views

Difference of binomial coefficients?

Let's say I have a sum of binomial coefficients that look like this: $12\choose5$+$11\choose5$+$10\choose5$+$9\choose5$+$8\choose5$ How can I rewrite this equation so that it's a difference of ...
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151 views

Orthogonality relation as double sum of products of binomial coefficients

I have stumbled upon the following sum over $x,y$ for non-negative integers $\kappa,\lambda$: $$ \sum_{x=0}^{\kappa}\sum_{y=0}^{\lambda}\left(-\right)^{x+y}{2\kappa \choose 2x}{2\lambda \choose 2y}\...
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28 views

coefficient of a term in an expansion

The coefficient of $x^{26}$ in expansion of $(1+x)^{41}(1-x+x^2)^{40}$ is ? Answer is $2082$ now on simplifying i get it as $(1+x^3)^{40}.(1+x)$ now this nowhere gives any coefficient with x to power ...
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0answers
25 views

Pascal triangle with non equiprobable events

As many of you will know when flipping n times a coin where each side is equally probable we can calculate the probability of getting x times heads with the triangle of pascal, that would be ${n \...
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42 views

Showing that percentage of the coefficients of $(x+y)^n$ being even tends towards $100\%$ when taking the limit of $n$ to infinity

A couple of weeks ago I came up with this function which could determine the number of coefficients divisible by some number $m$ in the binomial expansion of the expression $(x+y)^q$: iff $$p=\sum_{...
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49 views

Summation of this converging series [closed]

What's the sum of $$1+\frac{1}{3}.\frac{1}{2}+\frac{2}{3}\frac{5}{6}\frac{1}{2^2}+\frac{1\cdot2\cdot5\cdot8}{3\cdot6\cdot9\cdot2^3}+\cdots$$ I think it's the expansion of some expression but can't ...
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72 views

sum of multiplication of two binomial coefficients

Is there any formula for calculating $\sum_{k=0}^n {n\choose k} {2n\choose 2k}$ ? One possible way is to use Stirling's approximation, but couldn't reach a reasonable answer.
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1answer
59 views

An urn contains $8$ red balls and $12$ black balls and $10$ are removed at random?

An urn contains $8$ red balls and $12$ black balls and $10$ are removed at random. Find the probability that $7$ black balls are removed. Hint: Use binomials to count the number of ways to get $7$ ...
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112 views

A sum involving binomial coefficients and powers of 2

I am interested in a simplified version of the following sum $$\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^k}{2^k-1}.$$ I have to evaluate it for values of n ranging from $10^{4}$ till $10^{10}.$ Is there a ...
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3answers
106 views

Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.

Prove that $\binom{n+0}0 + \binom{n+1}1 +\binom{n+2}2 +\ldots+\binom{n+r}r = \binom{n+r+1}r$ using combinatoric arguments. (EDITED) I want to see if I understood Brian M. Scott's approach so I ...
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2answers
41 views

$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ where $a_r=\binom{20}{r}$, find the value $\mathop{\sum\sum}_{0\le i<j\le 20}(a_i-a_j)^2$

$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ where $a_r=\binom{20}{r}$, find the value $$\mathop{\sum\sum}_{0\le i<j\le 20}(a_i-a_j)^2$$ I first calculated the value of $\mathop{\sum\sum}_{0\le i<j\le ...
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2answers
36 views

Limit of sum of terms containing binomial coefficients

$$\lim_{n \to \infty} \sum_{k=0}^n \frac{n \choose k}{k2^n+n}$$ The result is $0$. The $n$ from the denominator can be ignored. If not for the $k$ at the denominator, the result would be $1$, but I ...
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3answers
144 views

Finite Double Sum $\sum_{j=0}^n\sum_{i=0}^j \binom {n+1}{j+1}\binom ni =2^{2n}$

The problem is given in a combinatorics class study sheet. I cannot prove, and actually I am not sure if there was a mistake in the question or not. I tried for a few small n's e.g. 1, 2 and it holds. ...
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3answers
204 views

Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$

I want to prove the following equality: $$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$$ So I had an idea to use $((1+x)^n)'=n(1+x)^{n-1}$ So I could just use the binomial theorem and ...
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1answer
40 views

If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r $ then find the required value

If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r $ and $(1+x+x^2)^{s}=\sum_{r=0}^{2s} b_rx^r$, then value of $\frac{\sum_{s=0}^{n}\sum_{r=0}^{2s} b_r}{\sum_{r=0}^{2n+1} \frac{a_r}{r+1}}$ will be: (A) ...
6
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2answers
114 views

How many integer-sided right triangles are there whose sides are combinations?

How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$? Attempt: This seems like a ...
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3answers
99 views

Number of terms in the expansion of $\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n$

Number of terms in the expansion of $$\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n$$ $\bf{My\; Try::}$ We can write $$\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n=\frac{1}{x^{2n}}\left(1+x+x^2\right)^n$...
8
votes
4answers
411 views

Finding coefficient of polynomial?

The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + …)^3$ is_______? Somewhere it explain as: The expression can be re-written as: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$ ...
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0answers
35 views

Bounding the summation of binomial terms

For $0<\theta<1, \theta'=(1-\epsilon)\theta, \epsilon< \theta, k\in\mathbb{N}$, the problem is to tightly upper bound the following binomial summation: $$\sum_{i=\lceil \theta k \rceil}^k {...
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2answers
48 views

Representing geometric series as sum of binomial coefficients

I was learning generating functions, where faced following: $$(1-x)^{-n}=\sum_{i=0}^\infty\binom{n+i-1}{i}x^i$$ I didn't get how is this arrived. I tried out to prove this to me. But failed. ...