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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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 ...
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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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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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61 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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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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81 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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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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118 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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28 views

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} ...
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159 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: $$ ...
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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... ...
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68 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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143 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 ...
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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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1answer
41 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 ...
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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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68 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
57 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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105 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
97 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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$(1+x)^{20}=\sum_{r=1}^{20}a_rx^r$ 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 ...
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35 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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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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193 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) ...
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113 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 ...
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91 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 ...
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404 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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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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46 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. ...
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186 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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190 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 ...
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36 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 ...
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1answer
89 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 ...
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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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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
29 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 ...
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1answer
75 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
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3answers
51 views

Simplify triangular sum of triangular numbers: $\sum_{i=1}^{n}(\frac12i(i+1))$

I'd like to simplify this expression, which sums up the first $n$ triangular numbers: $$\sum_{i=1}^{n}(\frac12i(i+1))$$ which is equal to: $$\sum_{i=0}^{n}((n-i)(i+1))$$ Is it even possible without ...
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1answer
56 views

Prove that $\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^m=…$ [closed]

Let $\binom{n}{k}$ denotes the number of subsets with $k$ elements in $n$-elements set. Prove that $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^m=\begin{cases} 0, & \text{ if } 0\le m \le n-1; \\ 2^n ...
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1answer
57 views

find a value in pascal triangle given row and column

How can I find a value from this pascal triangle given row and column number without calculating $^nC_r$? For example, for row=$4$, column=$3$: value is $10$, For row=$3$, column=$5$: value is ...
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1answer
35 views

How to get from left to right-hand side of the equation? $ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $

I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side. $$ ...
4
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1answer
65 views

Product of sums into a sum of products

Any idea on how I can get an expression in the form of sum of products from the following one?: \begin{equation} \prod_{i=1}^M \left(\sum_{n=1}^i x_n\right) \end{equation}
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2answers
41 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...