# 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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### Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
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### Number of draws that contain at least $k$ red coins

Assume there are $n$ coins in an urn from which $r$ are read. What is the number of draws of $r$ coins that contain at least $k$ red coins? It is obvious that there are $$\binom{r}{k}\binom{n-r}{r-k}$$...
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### Limit of binomial coefficients

Let $0\leq a_n\leq n$ be a sequence of integers. Under which condition on the $a_n$ does $$\frac{{n-a_n\choose a_n}}{{n\choose a_n}}=\frac{(n-a_n)(n-a_n-1)\dots(n-2a_n+1)}{n(n-1)\dots(n-a_n+1)}$$ (...
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### highest power of prime $p$ dividing $\binom{m+n}{n}$

How to prove the theorem stated here. Theorem. (Kummer, 1854) Let $p$ be a prime. The highest power of $p$ that divides the binomial coefficient $\binom{m+n}{n}$ is equal to the number of "carries"...
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### Coefficient of polynomials

Could someone explain to me why $$[x^{24}](1-2x^6)^{-31} = 2^4 \binom{4 + 31 - 1}{31 - 1} \, ?$$ Reads: The coefficient of $x^{24}$ in $(1-2x^6)^{-31} =$ ...
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### Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) [duplicate]

Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$$ holds?
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### What's the interpretation of $\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$?

I'm having problems with finding the combinatorial interpretation of this sum: $$\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$$ Can anyone help, please?
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### How prove $\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$

we know Bernoulli number such identity $$\sum_{k=0}^{n}\binom{n+1}{k}B_{k}=0$$ see:Bernoulli number identity show that $$\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$$ where $B_{n}$ ...
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### Generating function of binomial coefficients ${n\choose5}$

How to prove easily this identity for (almost classical) series with binomial coefficients: $$\sum_{n=5}^\infty \dfrac{\binom{n}{5}}{2^{n+1}} = 1 .$$ Thank you. Any smart proof would be much ...
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### Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
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### How find this sum $\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$

Let $m$ is give postive integer numbers, Find the sum $$\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$$ if this not have closed form,and can you use Special function ?
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### How to simplify $F(x)=\sum_{n}^{\infty}\sum_{k}^{\infty}{n-k-1\choose k}x^n$?

This generating function is equivalent to $$\sum_{n}^{\infty}F_n x^n=\dfrac{x}{1-x-x^2}$$ where $F_n$ is a fibonacci number. To show this, I need to simplify the above generating function with ...
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### Recurrence relation for product of binomial coefficients

We all know the standard recurrence relation for binomial coefficients: $$\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$$ Is there any finite-step recurrence relation one can write down for a ...
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### Combinatorial proof involving reciprocals

This is a follow-up to this question: show that if $n$ is a positive integer then $$\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}\binom{n}{k} =\sum_{k=1}^{n}\frac{1}{k}\ .$$ I was able to answer the question by ...
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### convergence of a sum of binomials

how can I show this converges to zero for some constant C as large as you need? $$\lim\limits_{n\rightarrow\infty} \sum\limits_{k=C\sqrt{ n\log(n)}}^{n}{n \choose k } 2^{-n}$$
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### How to calculate the sum of binomials? [closed]

I want to prove below: n is natural number. $$\sum_{k=1}^n k \binom{2n}{n+k} =\frac{1}{2}(n+1) \binom{2n}{n+1}$$ Please tell me above proof.
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### Binomial representation

We already know that we can represent this binomial as the following: $$(a+b)^K=\sum _{n=0}^K \binom{K}{n} b^n a^{K-n};$$ where $\binom{K}{n} = \frac{K!}{n! (K-n)!}$ I want to know if this ...
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### Binomial coefficient proof

Prove that for any $0\lt r\lt n$ we have $$\binom nr=\binom{n-1}{r-1}+\binom{n-1}r.$$ How do prove this and what step do i take in order for it to be true?