# Tagged Questions

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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### Closed formula for ${r \choose 1}+{r \choose 2}\cdots{r \choose w}$ where $w < r$ [on hold]

Let $r,w \in \mathbb{N}$. Are there some formula for the next sum? $${r \choose 1}+{r \choose 2}\cdots{r \choose w}$$ where $w<r$?
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### Combinatorial identity's algebraic proof without induction. [duplicate]

How would you prove this combinatorial idenetity algebraically without induction? $$\sum_{k=0}^n { x+k \choose k} = { x+n+1\choose n }$$ Thanks.
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### Finite summation including binomial coefficients and double factorials

I came across the following summation: $$\sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}).$$ $\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$. Mathematica told ...
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### Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
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### A sum involving binomial coefficients and its evaluation using the Gamma function [duplicate]

Does anyone know how to prove (or a reference for) the following identity for positive integers $r$: $$\sum_{i=0}^r (-1)^i{r\choose i}\frac{1}{ir+1}= \frac{\Gamma(1+1/r)\Gamma(r+1)}{\Gamma(r+1+1/r)}$$
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### Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $[q:1,1,1,1..1,2,2,..2]$. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
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### Given $n$ heads out of $n$ tosses. What is the posterior probability that coin is fair? [closed]

I am given an $\sigma$-fair coin with the probability of head $(\theta)$ being in the interval $[\frac{1}{2} - \sigma, \frac{1}{2} + \sigma]$. Also I am given: For a Bayesian analysis of the ...
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### Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion

If $l$ and $n$ are any positive integers, is there a proof of the identity $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!\;$$ which uses the Inclusion-Exclusion Principle? (If necessary, ...
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### Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. Here's the formula: $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$ Can anyone give a proof of this result? Note:...
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### Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
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### Relation between different ways of accessing bernoulli numbers

Bernoulli numbers can easily be expressed by linear algebra equations. For example just using the recursion formula $$\sum_{k=0}^{n-1}{n\choose k}B_k=0$$ which is equation (34) from the MathWorld ...
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### Sum of sum of binomial coefficients $\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
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### Asymptotics of $f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k}$

Define $$f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k}$$ for some fixed constant $c$ (say, $0<c<1/2$). What are the asymptotics of $f_c(n)$ as $n\to\infty$? It seems that this should be ...
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### Sum of reciprocals of binomial coefficients: $\sum\limits_{k=0}^{n-1}\dfrac{1}{\binom{n}{k}(n-k)}$

I'm trying to find a closed solution to the following binomial sum, without success. I would appreciate any assistance towards a solution. $$\sum\limits_{k=0}^{n-1}\dfrac{1}{\binom{n}{k}(n-k)}$$ ...
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### Sum of products of binomial coefficients: $\sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j}$

In a proof I've come across the following identity: $$\sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j}$$ I see that it's right, when plugging in numbers, but I don't see ...
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### Combinatorial Identity with Binomial Coefficients: ${{a+b+c-1}\choose c} = \sum_{i+j=c} {{a+i-1}\choose i}{{b+j-1}\choose j}$

I got the following identity from commutative algebra. I am curious to see elegant elementary methods. $${{a+b+c-1}\choose c} = \sum_{i+j=c} {{a+i-1}\choose i}{{b+j-1}\choose j}$$
### Find a binomial coefficient equal to ${n\choose k} + 3 {n\choose k-1} + 3{n \choose k-2} + {n\choose k-3}$
Exercise. Find a binomial coefficient equal to: $${n\choose k} + 3 {n\choose k-1} + 3{n \choose k-2} + {n\choose k-3}.$$ I don't really understand what we are asked to do when we are told to ...