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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Bounding simple series involving binomial coefficient

Let $r \ge 1$. What is a simple argument to show the following two inequalities: \begin{align*} \sum_{m=1}^n 2^m \binom{n}{m}^2 \Big( \frac{en}{m}\Big)^{-5rm} &\le n^{-r} \\ \sum_{m=1}^n ...
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How can I find the coefficient of x when the power is greater than the powers of 2 brackets using binomial expansion?

I have been given this question: Find the coefficient of $x^{13}$ in the expansion of $(1 + 2x)^4(2 + x)^{10}$. I know how I would find $x^4$ or lower degrees, but I am unsure how to approach this, ...
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2answers
102 views

How to prove $\sum_{i=1}^ki^k(-1)^{k-i}\binom {k+1}{i} =(k+1)^k$

How to prove $\sum_{i=1}^ki^k(-1)^{k-i}\binom {k+1}{i} =(k+1)^k$ where k is a positive integer. Any hints can help.
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3answers
105 views

Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$

I've worked out a proof, but I was wondering about alternate, possibly more elegant ways to prove the statement. This is my (hopefully correct) proof: Starting from the identity $2^m = \sum_{k=0}^m ...
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Problem with induction of binomial coefficiency

(Sorry for making up math language, I am roughly translating math terms here) This is part of some of the induction exercises in the book "Otto Forster: Analysis 1" (1.2): ...
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1answer
38 views

Can this be proved using definite integrals [duplicate]

It's a problem from a high school math book that I've been unable to solve: Prove using definite integrals that, $${n \choose 1}-\frac{1}{2}{n \choose 2}+\frac{1}{3}{n \choose ...
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1answer
40 views

How to show $\phantom{d}_d C_0+\phantom{d}_d C_4 + \cdots = 2^{d-2} + 2^{\frac{d}{2}-1} \cos(\frac{d \pi}{4}) $?

I want to show following identities \begin{align} &\phantom{d}_d C_0+\phantom{d}_d C_4 + \cdots = 2^{d-2} + 2^{\frac{d}{2}-1} \cos(\frac{d \pi}{4}) \\ &\phantom{d}_d C_1+\phantom{d}_d C_5 ...
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2answers
33 views

Find $\lim_{n\to\infty}(\frac{\sideset{^{3n}}{_n} C}{\sideset{^{2n}}{_n} C})^{\frac{1}{n}}$

Find $\lim_{n\to\infty}(\frac{\sideset{^{3n}}{_n} C}{\sideset{^{2n}}{_n} C})^{\frac{1}{n}}$ Let $L=\lim_{n\to\infty}(\frac{\sideset{^{3n}}{_n} C}{\sideset{^{2n}}{_n} C})^{\frac{1}{n}}$ $\log ...
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2answers
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Binomial Coefficient Inequality

I don't know how to prove this inequality $$ \binom{n}{0} < \binom{n}{1} < \binom{n}{2}< ... <\binom{n}{\left \lfloor {\frac{n}{2}}\right \rfloor} $$ Knowing that $$ ...
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1answer
37 views

Cardinality of the set $\{(X_1,X_2,\cdots,X_p)\in \mathcal{P}(E)^p \mid X_1\subset\cdots\subset X_p \}$

If $|E|=n$ is a set, what is the cardinality of the set $$\{(X_1,X_2,\cdots,X_p)\in \mathcal{P}(E)^p \mid X_1\subset\cdots\subset X_p \}$$ My thoughts The giving of p-tuplets ...
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3answers
88 views

The greatest integer less than or equal to the number $R=(8+3\sqrt{7})^{20}$

Given $$R=(8+3\sqrt{7})^{20}, $$ if $\lfloor R \rfloor$ is Greatest integer less than or equal to $R$, then which of the following option(s) is/are true? $\lfloor R \rfloor$ is an even ...
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2answers
60 views

What is the $x^8$ term in $(x+\frac{1}{\sqrt[3]{x}})^5(2x - \frac{1}{x^2})^7$?

What is the $x^8$ term in $$(x+\frac{1}{\sqrt[3]{x}})^5(2x - \frac{1}{x^2})^7$$? I know how to expand $(a+b)^n$ via Newtons binome, but I can't find an elegant way of determining the $x^8$ term ...
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1answer
50 views

Sum of Series of Binomial Coefficients.

Find the sum of $\binom{200k}{0}+\binom{200k}{100}+\binom{200k}{200}+...+\binom{200k}{200k}$ and $\binom{200k}{1}+\binom{200k}{101}+\binom{200k}{201}+...+\binom{200k}{200k-99}$ in terms of $k$. I've ...
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For what values of $n$ do more than two thirds of subsets of $[n]$ have size beteen $\frac{n}{3}$ and $\frac{2n}{3}$

My actual problem is slightly different. For what values of $n$ do we have: $2\sum_{i=1}^{\lfloor n/3 \rfloor}\binom{n}{i}\geq \sum\limits_{i=\lfloor n/3 \rfloor+1}^{\lceil n/2 \rceil-1} ...
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2answers
106 views

Finding the infinite series: $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$

Evaluating $$\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!n!}{(m+n+2)!}$$ involving binomial coefficients. My attempt: $$\frac{1}{(m+1)(n+1)}\sum_{m=0}^\infty ...
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1answer
55 views

Supercongruence for Binomial Coefficients

$${p^{e+1} \choose p\cdot k } \equiv {p^{e} \choose k } \mod p^{e+1} $$ $p$ is prime, $e$ and $k$ are non negative integers. I am struggling with a proof of the above proposition, in the ...
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1answer
81 views

Proving Singmaster's Conjecture: Can you prove there are finitely many solutions of $\binom{n+x-1}{n}=y$?

$n,x,y \in\mathbb{N}, \binom{n+x-1}{n}=y$ I am completely out of my depth. What I am really asking is how can you prove there is an upper bound on the number of pairs (n, x) to satisfy an arbitrary ...
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1answer
86 views

Binomial sum gives $4^n$

I was looking at this question:Swapping the $i$th largest card between $2$ hands of cards and WolframAlpha gave me this result. Why is it so? $$\sum_{k=0}^n{2k\choose k}{2n-2k\choose n-k}=4^n?$$
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Is there a name for this discrete version of Jensen, specifically when applied to binomial coefficients?

We have $2k$ integers greater than or equal to $j\geq0$ $a_1+a_2+\dots + a_k=n$ and $b_1+b_2+\dots + b_k=n$. If for all $1\leq i\leq k$ we have $|n/k-a_i|\leq|n/k-b_i|$. Then ...
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Find the minimal $n$ between $1$ and $2015$ such that $\binom{2015}{n}$ is even

I would appreciate if somebody could help me with the following problem Q: Find minimum $n(n\in \{1,2,3,4,\cdots,2015\})$, following holds $$\binom{2015}{n}: \text{is even number}$$
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2answers
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Sum of binomial coefficients $\sum_{n=0}^6 \binom{6}{n} = 2^6$

I assume this is a rather simple result, but I am not sure how to arrive at it. Apparently: $$\sum_{n=0}^6 \binom{6}{n} = 2^6$$ I can sum over all the binomial coefficients and verify this of ...
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1answer
33 views

Proof of Binomial Coefficients Comparison Inequality

Please help to prove the inequality $$ \binom{a}{b}\leq\binom{a+j}{b+i}$$ For $i\leq j$ Using the basic identity $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ I have proceeded to $$ 1 \leq ...
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1answer
47 views

Hockey-Stick Theorem for Multinomial Coefficients

Pascal's triangle has this famous hockey stick identity. $$ \binom{n+k+1}{k}=\sum_{j=0}^k \binom{n+j}{j}$$ Wonder what would be the form for multinomial coefficients?
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Proof of inequality involving binomial coefficients

Proving things is not my forte. Stumbled upon following identity: $$\binom{n+k+1}{k}>\sum_{i=1}^k \binom{\alpha_i}{i}$$ For $\alpha_i<\alpha_j $ for $i<j$ $0\leq \alpha_i \leq n+k$ Also ...
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1answer
73 views

Is there a closed form for $n^k$ in terms of $\Delta n^{k+1},\Delta n^k$, …?

Let $\Delta$ be a sort of difference operator on a function $f(n)$ such that $$\Delta f(n)=f(n+1)-f(n)$$ Take the basic power function $f(n)=n^k$, $k\in\mathbb{N}\cup\{0\}$. Then we get ...
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Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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176 views

Curious combinatorial summation

Let $\gamma$ denote a grid walk from the upper left corner $(1,k)$ to the lower right corner $(\ell,1)$ of the $k\times\ell$ rectangle $\{1,..,k\}\times\{1,..,\ell\}$. There are ...
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Product of two binomial coefficients in terms of linear combinations of binomial coefficients [duplicate]

For $n,a,b$ natural numbers with $a+b \leq n$, can we find positive rational numbers $c_1, \ldots, c_{a+b}$ such that $$ {n \choose a}{n \choose b} = \sum_{k=1}^{a+b}c_k{n \choose k}? $$ Is there a ...
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Prove that $\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{m+k}{r}=0$

It is well-known that $\sum_{k=0}^n(-1)^k\binom{n}{k}=(1-1)^n=0$. It is seems like that $$\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{m+k}{r}=0$$ for any $m,r\in\mathbb{N}$, $r\leq m$. How to prove or ...
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1answer
61 views

Infinite summation of reciprocal of Binomial coefficients [duplicate]

When I was playing with Binomial coefficients. I got an interesting problem. It is very nice formula. The problem is $$\large \sum_{n=0}^{\infty} \dfrac{1}{\binom{2n}{n}}$$ where $\binom{2n}{n}$ ...
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Prove that : $\sqrt C_1+\sqrt C_2 +\sqrt C_3 \ … +\sqrt C_n \leq \sqrt{n(2^n-1)}$

If $ C_0, C_1 , C_2, ... , C_n$ are the combinatorial coefficients in the expansion of $(1 +x)^n$, $n\in N$, then prove the following :$$\sqrt C_1+\sqrt C_2 +\sqrt C_3 \ ... +\sqrt C_n \leq ...
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108 views

A sum related to binomial theorem

If $\dfrac{x^2+x+1}{1-x} = a_0+a_1x+a_2x^2+\cdots$ then $\displaystyle\sum_{\gamma = 1}^{50}a_{\gamma} = ??$ Original Image This is a sum related to evaluating a series, from the chapter ...
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103 views

A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}. $$ Recently, I have found ...
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1answer
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How do you evaluate this summation: $S=\sum\limits_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$

Find S: $$S=\sum_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$$ My attempt: I tried writing the summation as: $$S=3!(15!)\sum_{r=0}^{15} (-1)^r \frac{1}{(15-r)!(r+3)!}$$ and tried to ...
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An interesting Sum involving Binomial Coefficients

How would you evaluate $$\sum _{ k=1 }^{ n } k\left( \begin{matrix} 2n \\ n+k \end{matrix} \right) $$ I tried using Vandermonde identity but I can't seem to nail it down.
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1answer
69 views

generalized binomial coefficient

I know that the coefficient $$-\frac{1}{2} \choose k$$ can be simplified by multiplying both the nominator and the denominator by $$2^k$$ and then represented as $$ (-\frac{1}{4})^k {2k\choose k}$$ ...
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Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
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1answer
46 views

Sum of kth binomial coefficient but starting at index 1.

I want to know what is the sum $\binom{n}{1} + \binom{n}{1+k} + \binom{n}{1+2k} +...$. The answer for when it starts at 0 can be derived with roots of unity and simply adding, and I suspect the same ...
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69 views

Prove this binomial sum [closed]

Following problem is interesting Show that: $$\sum_{i=1}^{n-1}\binom{n-1}{i} i^{i-1}(n-i)^{n-i-1}=n^{n-1}-n^{n-2}$$
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2answers
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Subsets and binomial coefficients

Assume that $R$ is a set with $n$ elements. We know that the number of subsets of $R = 2^n$. What does this statement have to do with the binomial coefficient?
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1answer
79 views

How to find the sum $\sum_{k=0}^{n}\binom{n+k}{k}$?

How to find the sum $\displaystyle\sum_{k=0}^{n}\dbinom{n+k}{k}$? In my book it is written as a hint that we can use the formula $\dbinom{n}{r}+\dbinom{n}{r-1}=\dbinom{n+1}{r}$ to find it.But I can't ...
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1answer
86 views

Series with a reciprocal of the central binomial coefficient

How can we prove the following identities $$\sum_{n=1}^\infty n^{-3}{\binom{2n}n}^{-1}=\pi\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)-\frac{4}{3}\zeta(3)\tag{1}$$ $$\sum_{n=1}^\infty ...
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2answers
65 views

If $n$ is a product of primes, what is the number of divisors?

Let $n=p_1p_2...p_k$ Then the number of divisors is what? I assumed it was $1+ \binom k1+ \binom k2 + \binom k3 + ... + \binom kk=2^k$ Is this correct? Prove that the number of divisors is odd ...
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2answers
63 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
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1answer
106 views

Number of permuatations such that no two consecutive letters are neither vowels nor identical

Find number of different permutations of all the letters of the word "PERMUTATION" such that any two consecutive letters are neither both vowels nor both identical. The vowels in this word are ...
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1answer
26 views

Binomial coefficients sum convergens

Find when $\sum_{n=1}^{\infty }\binom{\alpha}{n}$ ($\alpha$ is a real number) diverges, converges, or converges absolutely. First I notice that it is basically ${(1+1)}^{\alpha}$ so the sum ...
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1answer
47 views

Prove using the binomial theorem that $(1+\frac{1}{n})^n < \sum_{j=0}^n \frac{1}{j!} < 2 + \frac{1}{2} + \frac{1}{4} + …+ \frac{1}{2^{n-1}}$

I understand how to prove this problem, essentially the middle term $\sum_{j=0}^n \frac{1}{j!}$ is equal to the Euler's number, e, and the third term in this sequence is equal to 3. However, I am not ...
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1answer
52 views

Proving an inequality that looks related to the Binomial series,

Edit: I changed the inequality to the one that I think was meant to be asked. This is a former exam question from my math dept, and it is relatively old - from 1993. So, I think there was a typo on ...
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2answers
64 views

How to make sense of the binomial coefficient over $p$-adic integers?

I recently asked this question: Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$. and now I'm trying to make sense of the first answer that was posted. It said that I should show ...
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3answers
483 views

After 6n roll of dice, what is the probability each face was rolled exactly n times?

This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible ...