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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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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34 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
16 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
22 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
46 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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1answer
37 views

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

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
35 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
34 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. $$ ...
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1answer
57 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
38 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. ...
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5answers
66 views

Give a proof of ${n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + … + {n \choose n}^2 = {2n \choose n}$ [duplicate]

I must prove this: ${n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + ... + {n \choose n}^2 = {2n \choose n}$ But, I have no idea how to prove it or how it necessarily works. Could someone help ...
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1answer
41 views

What is the simplest way to show that ${(p-1)! \over (k)!(p-k)!}$ is an integer?

In the proof of $p$ | $\binom{p}{k}$ (p divides $\binom{p}{k}$) where $p$ is prime, what is the simplest way to show that $${(p-1)! \over (k)!(p-k)!}$$ is an integer?
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1answer
31 views

Use induction and Pascal's Identity to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$

I know Pascal's Identity is ${n \choose k}={n-1 \choose k-1}{n-1 \choose k}$, but I am not sure how to set up and use the proof to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$. Can anyone help me ...
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2answers
37 views

Give a formula for the coefficient of $x^{k}$ in the expansion of $(x+{1 \over x})^n$

I followed the binomial theorem and got this: The Binomial Theorem is: $(a+b)^{n}= \sum_{k=0}^{n} {n \choose k}{a}^{k}{b}^{n-k}$ Then let $a=x, b={1\over x}, n = n, k = k.$ I then get ...
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5answers
155 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
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1answer
17 views

Solving integration equation with binomial expression using gamma function [closed]

Can anyone help me how they have solved this? Denote $\beta_1 = (\alpha-1)/2$ and $\beta_2 = (\alpha+1)/2$. The total power received at the BS is $$P_{bs}^{tot} = \int_{-\infty}^\infty ...
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1answer
35 views

What is this matrix notation and how is it solved?

I've never taken a stats class, or linear algebra or much of anything that involves matrices. In one of my books they give me this as part of an example and it states, $$\binom{6}{4} = 15 \text{ ...
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3answers
76 views

Is there an expression for the sum of $\binom nr^2$ for each $n$? [duplicate]

Is there a standard expression for $$\sum_{r=0}^{n}\binom nr^2$$
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177 views

summation of a binomial expression that doesn't start from 0

I have the following expression: $$ \sum_{k=9}^{17}\binom{17}{k} $$ and I need to show that it's equal to: $$ 2^{16} $$ now I know that if 'k' was starting from zero and not from 9 , like this: $$ ...
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1answer
35 views

Evaluation of all positive integer ordered pair $(n,r)$ for which $\displaystyle \binom{n}{r} = 2016$

$(1)$ Evaluation of all positive integer ordered pair $(n,r)$ for which $\displaystyle \binom{n}{r} = 120$ $(2)$ Evaluation of all positive integer ordered pair $(n,r)$ for which ...
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0answers
29 views

Simplify binomial coefficients sum [duplicate]

Exercise requires to simplify this sum: $$\sum_{k=0}^{20} \binom{50}{k}\binom{50}{20-k}$$ Tried to figure this out with no success. I have only final answer, which is $\binom{100}{20}$. Please help ...
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2answers
40 views

Simplify the sum of binomial coefficients

The exercise requires to simplify the following expression: $$\sum_{k=0}^{25} \binom{50}{2k}$$ By finally looking at someone's answer, I know that the result should be $2^{49}$, but the following ...
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1answer
24 views

Binomial distribution, explanation formula

I have a really simple question. I can't figure out the meaning of the binomial coefficient in the case of a binomial distribution formula. I know what the formula means, and how to use it for the ...
3
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3answers
54 views

how to come up with this identity $\sum\limits_{i=r}^{n-k+r}{i \choose r}{{n-i} \choose {k-r}}={{n+1} \choose {k+1}}$

This identity is used in an exercise. Could you help me understand how I should reason to come up with it? Ideally, from a combinatorial point of view.
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1answer
47 views

Prove that $\frac{(2n)!}{(n!)^2}-1$ is divisible by $(2n+1)$

Prove that $$\frac{(2n)!}{(n!)^2}-1$$ is divisible by $(2n+1)\;,$ Where $n\in \mathbb{N}$ and $n>1$ $\bf{My\; Try::}$ Let $$S = \frac{(2n)!}{n!^2}-1 = \frac{2^n(2n-1)(2n-3)\cdot \cdot ...
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1answer
48 views

Binominal expression simplification

I need to simplify the expression $$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$ Thank you.
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3answers
68 views

Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
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1answer
114 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
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1answer
28 views

A Binomial coefficient sequence

If 'n' is a positive integer and $C_k=^nC_k$, then find the value of: $[\sum\limits_{k=1}^n\frac{k^3}{n(n+1)^2.(n+2)}(\frac{C_k}{C_{k-1}})^2]^{-1}$ [![enter image description here][1]][1] I have ...
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37 views

Does the property ${n\choose r}={n\choose n-r}$ have a name?

Due to the relation between Pascal's Triangle and the choose function in probability theory, we can deduce that $${n\choose r}={n\choose n-r}$$ because Pascal's Triangle is symmetric. This can also be ...
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1answer
58 views

Closed form for a binomial identity [closed]

$\textrm {How do I find a closed form for } \sum_{j=0}^n{} j\displaystyle\binom{j}{r} = ?$ Is this some kind of upper index summation? Any previous papers? Thank you
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4answers
60 views

How many numbers are there of 2n digits that the sum of the digits in the first half equals the sum of the digits in the second half

The question is how many number of a given number of digits 2n where the sum of the first half of the digits equals the sum of the digits in the second half. So this is for a programming problem and ...
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0answers
32 views

Can someone expain to me what's going on (binomial coefficient)?

I'm watching this proof for $\zeta(2n)$ on YouTube. This is what I can understand so far: $${s\over e^{s} -1} = \sum^{\infty}_{n=0} {\beta_n\over n!} s^n$$ Where $\sum^{\infty}_{n=0} {\beta_n\over ...
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2answers
25 views

Induction with binomial coefficient

Is mathematical Induction possible with this sigma sign? $A(k) =\sum_{j=0}^{k} \binom{m}{j}\binom{n}{k-j} = \binom{m+n}{k}$ $A(k+1) = \sum_{j=0}^{k+1} \binom{m}{j}\binom{n}{(k+1)-j} = ...
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2answers
37 views

Bound on sum of combinations

I came across the following inequality $\sum_{i=0}^D \binom N i \le N^D+1$. I am not sure how to prove this. I tried to do it by induction on $D$, and started with observing the values of sum for ...
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1answer
53 views

Gaussian polynomial identities

I'd appreciate any hints for showing that these identities are true for Gaussian polynomials. I've tried to approach the problem using basic algebra but it gets messy very quickly and I've gotten ...
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1answer
79 views

Simplifying my sum which contains binomials

While dealing with compositions (ordered partitions) of integers, I found the following formula for the shifted $m$-generalized Fibonacci numbers (Wikipedia: Generalizations of Fibonacci numbers): ...
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3answers
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Reducing this binomial expression [closed]

I need help for showing that: $$\sum\limits_{k=2}^{50} = k \cdot(k-1)\binom{50}{k}$$ is equal to: $$50\cdot 49\cdot 2^{48}$$ please help , thank you.
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1answer
72 views

Prove that $\binom{n}{0}\cdot \binom{2n}{n}-\binom{n}{1}\cdot \binom{2n-2}{n}+\binom{n}{2}\cdot \binom{2n-4}{n}+… = 2^n$

Prove that $$\binom{n}{0}\cdot \binom{2n}{n}-\binom{n}{1}\cdot \binom{2n-2}{n}+\binom{n}{2}\cdot \binom{2n-4}{n}+........... = 2^n$$ $\bf{My\; Try::}$ Coefficient of $x^n$ in ...
4
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0answers
64 views

Dealing with a difficult sum of binomial coefficients, $\sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j} $

I am interested in finding an upper bound for the sum $$F(n)= \sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j}.$$ Ideally it should be possible to evaluate it exactly using some ...
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2answers
60 views

What is the coefficient of $x^4$ in the expansion of $\sqrt[3]{1+x}$

Here's what I tried: $$\sum_{n \ge0} {\frac{1}{3} \choose n} x^n= \sum_{n \ge0} = \frac{\frac{1}{3}!}{n!(n-\frac{1}{3})!}x^n=\sum_{n \ge0} \frac{(\frac{1}{3}-1)(\frac{1}{3}-2)\cdot ...
2
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2answers
77 views

Show that $a_n=\frac{n+1}{2n}a_{n-1}+1$

Show that $a_n=\frac{n+1}{2n}a_{n-1}+1$ given that: $a_n=1/{{n}\choose{0}}+1/{{n}\choose{1}}+...+1/{{n}\choose{n}}$ The hint says to consider when $n$ is even and odd. When $n=2k$ I get: ...
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1answer
58 views

Proof of product summation of binomial coefficients

when I try to proof the sum of two independent negative binomial distribution to be negative binomial, I end up with how to proof the following identity. I try the induction but after I rearrange the ...
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2answers
75 views

In how many ways can I split 151 different objects into 3 categories?

In how many ways can I split 151 different objects into 3 categories such that no category gets absolute majority? I figured that the answer should be: ...
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1answer
22 views

Negative binomial coefficient

For $r \geq 1$, $k \geq 0$ both integers, I wish to show that $$\binom{-r}{k}^{*}(-1)^{k} = \binom{r+k-1}{k}$$ (the negative binomial coefficient is the left one). By definition, ...
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1answer
39 views

Show that $\sum_{i=\lceil\alpha n\rceil}^n {n\choose i}\le 2^{nH(\alpha)}, \alpha\in(1/2,1]$

For $1/2<\alpha\le 1$ show that $$\sum_{i=\lceil\alpha n\rceil}^n {n\choose i}\le 2^{nH(\alpha)}$$ where $H(\alpha)=-\alpha\log_{2}\alpha - (1-\alpha)\log_2 (1-\alpha)$ is the entropy. ...
6
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0answers
149 views

Conjecture $\sum_{n=1}^\infty\frac{(1/2)_n(-1/6)_n}{n!(2/3)_n}H_n\overset{?}={\pi\over 6}+2\sqrt{3}\ln(1+\sqrt{3})-{7\over\sqrt{3}}\ln 2-6+2\sqrt{3}$

Sums involving harmonic numbers have old history and first examples of their evaluations go back to Euler. Lately there has been a lot of interest, both on this forum and among professional ...
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2answers
42 views

Equality of two binomial coefficient containing expressions

Why is $$ \begin{align} &\sum_{k=0}^n(-1)^k\left[\binom{n-k-1}{k}+\binom{n-k-1}{k-1}\right]2^{n-2k}\\ ...
0
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2answers
46 views

What is the multiplication of two sigmas?

Say we have two sigmas $\sum_{i=0}^n\dbinom{n}{i}x^i$ and $\sum_{i=0}^m\dbinom{m}{i}x^i$, what would be the resultant of the above? How do you, in general, multiply two sigmas?
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120 views

What are the “numerator” and “denominator” of binomial coefficients called?

Do the numbers $n$ and $k$ in the binomial coefficient $\binom nk$ have a name? For the fraction $\frac nk$ we would use numerator and denominator. But I have not seen some terminology for binomial ...