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 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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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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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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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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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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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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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
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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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1answer
63 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
84 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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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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2answers
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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120 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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1answer
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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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186 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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160 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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1answer
35 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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104 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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144 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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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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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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1answer
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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70 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
58 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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108 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
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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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$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ 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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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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142 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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199 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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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) ...
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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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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$...
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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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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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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. ...
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188 views

How to find a simplified expression for $\binom{1/2}{n}$? [closed]

How to find a simplified expression for this specific binomial coefficient? $$\binom{\frac{1}{2}}{n}$$
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191 views

upper bounding alternating binomial sums

So we know that $\large\sum_\limits{i=0}^t\dbinom{m}{i}\dbinom{n-m}{t-i}=\dbinom{n}{t}$ by a simple counting argument. Now is there any bound on the quantity $\large\sum_\limits{i=0}^t(-1)^i\dbinom{...
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37 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
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1answer
25 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
2
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1answer
91 views

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where $_2F_1$...
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41 views

For any given $k$, show that an integer $n$ can be represented as: $n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$

For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$. My ...
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0answers
21 views

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$ [duplicate]

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$. Typically to combinatorially prove something we need to show that the LHS indeed counts the same ...
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1answer
30 views

Using Pascal's formula to derive another formula

Use Pascal’s formula repeatedly to derive a formula for $\dbinom{n+3}{r}$ in terms of values of $\dbinom{n}{k}$ with $k \leq r.$ (Assume $n$ and $r$ are integers with $n\geq r \geq 3).$ I have a idea ...