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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Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime.

Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime. Here $v_p(k)$ denotes the largest $\alpha\in\mathbb Z_{\ge 0}$ s.t. $p^\alpha\mid k$. We have ...
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2answers
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Why does $\binom{n}{2} = \frac{n^2 - n }{2}$?

In a proof in Introduction to Algorithms, the book says $\binom{n}{2} \cdot \frac{1}{n^{2}} = \frac{n^2 - n }{2}\cdot \frac{1}{n^{2}}$, which implies $\binom{n}{2} = \frac{n^2 - n }{2}$. Why are ...
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67 views

How to simplify $\sum_{i=0}^{2n - d - 1} {n \choose i}$?

Is it possible and how could I simplify this sum into a formula who's quantity of operations is independent of n? $$ \sum_{i=0}^{2n - d - 1} {n \choose i} $$ ...
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1answer
42 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...
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1answer
68 views

Binomial Coefficients

I want to get ahead in my classes and learn Binomial Theorem ahead of time. What I know so far is that the formula below is the Binomial Coefficient: $\binom n k = \frac {n!} {(n-k)!k!}$ and ...
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54 views

Calculate the sum $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$.

I want to calculate this sum: $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$. I tried to use some differentation techniques, but they didn't work. Could you help me with this?
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Proving if it is prime

I'm quite lost on how to prove things, with the $n \choose k$ and proving. So the question is: Prove that $n \choose k$ is divisible by $n$ if $n$ is a prime number and $1 \le k\le n-1$ Like, how ...
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58 views

Recursion Formula Euler Numbers

I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$ Where $E_k$ are the Euler Numbers. The approach that I have taken ...
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43 views

A sum involving a ratio of two binomial factors.

Let $a\ge 0$, $a_1\ge 0$ ,$b \ge 0$ and $b_1\ge0$ be real numbers subject to $1+b+a_1-b_1-a >0$. Let $m$ be a positive integer. Then using methods similar to those in Another sum involving binomial ...
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21 views

A multivariate sum that yields a closed form expression

Let $d\ge 2$ be a an integer. Let $b_1,b_2,\cdots,b_d$ be positive integers. As a by product of certain calculations I have discovered that: \begin{equation} \sum\limits_{q_2=0}^{b_2} \cdots ...
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1answer
55 views

Calculate wining probability in a dart game

Suppose we're playing the following dart game: The player can play up to $T$ rounds. In each round of the game, the player first throw a black dart, and then a white dart. Each dart independently hit ...
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3answers
517 views

Find the Pascal's Limit [closed]

Let $P_{n}$ be the product of the numbers in row of Pascal's Triangle. Then evaluate $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}}$$
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282 views

Find the sum of coefficient of all the integral power of $x$ in the expansion of $\big(1 + 2\sqrt x\big)^{40}$? [closed]

While going through certain question online. This question took a lot of my time. Can anyone please help me with this question!!
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1answer
40 views

Complex analysis ~ Binomial theorem

Given the identity $ \binom {2n} {n} = \frac{1}{2\pi i} \int_{C_r} \frac{(1+z)^{2n}}{z^{n+1}}dz,$ with $C_r$ the unit circle, prove that $\forall n \in \mathbb{N}$: $\binom {2n} {n} \leq 4 ...
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1answer
25 views

prove $\mathscr P_k (A)$ has ${n \choose k} $ elements

I am working on this question If $A$ is a set and $k\in \mathbb {N}$, let $mathscr P_k$ be the set of all subsets of $A $ that have k elements. Prove $\mathscr P_k (A)$ has ${n \choose k} $ ...
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1answer
37 views

Simplifying $\binom{n}{k}$ / $\binom{n}{k-1}$

So the question is as follows: Simplify $$ \frac{\binom{n}{k}}{\binom{n}{k-1}}. $$ And this is what I got: $$\begin{align*} \frac{\binom{n}{k}}{\binom{n}{k-1}} &= ...
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0answers
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The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial ...
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38 views

Binomial Theorem Help

In my assignment I have the following question: Alan
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64 views

Sequence closed expression or others

What are closed expression or any other expression (involving integrals, specials functions...) for $\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$ where $t>0$ integer Thank you
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2answers
102 views

pascal triangle, getting to the sum of it

So we know that $$\binom{n}0+\binom{n}1+\ldots+\binom{n}n=2^n\;.$$ What about the following sum? $$\binom{n}n+\binom{n+1}n+\ldots+\binom{n+m}n\;?$$ (a) Identify several examples of this sum on ...
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How to show $u_n=\binom{n}{l}\Rightarrow s_n=\binom{n+1}{l+1}$?

Hello respected everyone. Before I ask my query, let us first define binomial coeffcients as follows: For $n, r\in \mathbb N$, we define $$\binom{n}{r}=\begin{cases} \frac{n!}{r!(n-r)!}~~~ ...
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Dealing with floor function in binomial coefficients

I'm trying to estimate $\binom{n}{\left \lfloor{\alpha n}\right \rfloor }$ asymptotically using Stirling's formula. However, I'm a little lost with what to do about the floor function here. In the ...
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Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases:

Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases: (a) $x,y,z,w \geq 0$ (b) $x,y,z,w > 0$ (c) $x>2, y>-2, z>0, w>-3$ I think I have an idea on how ...
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1answer
33 views

How is the Binomial coefficient simplified to a falling factorial?

I'm learning how to take the derivative of the binomial coefficient and found a blog post that was quite useful. However I am unclear as to how the first step bellow was simplified to the second step ...
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1answer
39 views

Trailing zeroes in binomial coefficient

I have a doubt regarding trailing zeroes in binomial coefficients... Question: How would you calculate the number of trailing zeroes in the binomial coefficient of ${n\choose r}$ upto values of ...
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1answer
79 views

Number of ways to combine two sets of values

Good day. I have an algorithm which iterates over a set of values S =[1 -1 0.5 -0.5] for a set of parameters in vector ...
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29 views

Asymptotic for binomial coefficients

Does there exist any asymptotic formula for binomial coefficients ${n \choose k}$ for large $n$ when $k$ is fixed?
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Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$ where $|z|<1$ and $k \geq 0$. I know The right hand side: \begin{align*} \frac{1}{(1-z)^{k+1}} & = ...
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1answer
57 views

Remove first element of sequence, compute cumulative sum, iterate

(This is related to my previous question General formula for iterated cumulative sum.) Consider the sequence $S_0$ consisting of ones: $$ 1,1,1,1,1,1,\ldots $$ Now remove the first element and ...
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1answer
28 views

Integer Series Expansion

For any two $p,q \in \mathbb{Z}$ and $n \in \mathbb{Z}^+$, can one prove that $a_n = \frac{p(-p)^n - q(p-2q)^n}{(p-q)}$ is an integer with recursion relation $a_0 = 1,$ $a_n = ...
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How to computer the summation of a binomial coefficient/ show the following is true

$\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$. I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make ...
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1answer
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Show that $\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$

The Problem Show that $$\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$$ My Work $$\begin{align*}\frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k} &= ...
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34 views

How to show C(n,k)= C(n,n-k)

I am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the ...
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1answer
23 views

Binomial series after using binomial identity

Following http://en.wikipedia.org/wiki/Binomial_coefficient#Newton.27s_binomial_series , I am trying to prove that $$ \sum_{\kappa=0}^\infty \binom{\eta + \kappa}{\kappa}x^\kappa = (1 - ...
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Even/Odd Binomial Coefficients

I was wondering if there's a nice general solution for the following problem: How many numbers in the $n^\text{th}$ row of Pascal's triangle are even? How many are odd?
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1answer
225 views

Counting elements in cartesian power with plurality + pattern constraints

Problem: I have an alphabet with n=8 letters (say $X=\{A, B, C, D, E, F, G, H\}$). I'm looking for words with m=24 letters, with three constraints: letter $A$ is the relative majority (like in ...
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Prove Binom Sum $\sum_{k=0}^n(-1)^k \binom{n}{k} = 0$ [duplicate]

Let: $$ (-1)^0=1 $$ I need to prove that: $$ \sum_{k=0}^n(-1)^k \binom{n}{k} = 0 $$ Thanks!
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2answers
45 views

Differentiation of a binomial-like sum with respect to m (total number of trials)?

I want to differentiate a function with respect to m, and I know that I have to first find a representation for the whole sum to do that since m is discrete. However, I could not find one yet. Any ...
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Recovering the coefficients $b_r$ of the binomial sum $\sum_{r=0}^n\binom{n}rb_r$ [closed]

Suppose that the sequences of real numbers $a_0,a_1,a_2,a_3,\ldots$ and $b_0,b_1,b_2,b_3,\ldots$ satisfy the relation $$a_n=\sum_{r=0}^n\binom{n}rb_r\;.$$ Then prove that ...
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Can a certain polynomial have all its coefficients in some basis divisible by a prime $p$?

I fix $n\in\mathbf{N}^{*}$ and $n$ elements $\alpha_1,\ldots,\alpha_n$ in $\mathbf{N}^{*}$. Consider the polynomial $$Q(T)=\prod\limits_{1\leq i \leq n} \prod\limits_{0\leq j \leq \alpha_i -1} ...
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Negative binomial theorem

I have been supplied with a combinatorical proof based on the n'th power, however I am trying to prove this by induction. I have no problem with the base case, or assuming that n=N. However, for ...
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closed form for $\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$?

Does there exist any closed form for the following sum? $$\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$$ Edit: Then can we find an asymptotic nice approximation as $n\to ...
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1answer
44 views

Sum of binomial coefficients and powers

The following identity is true for $n\geq1$: $$ n!=\sum_{k=1}^n (-1)^{n-k} {n\choose k} k^{n} $$ You can obtain it from the equation in this question by setting the variables equal to 1. I was ...
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Induction proof: sum of binomial coefficients

Prove by mathematical induction for all natural numbers $n$: $$\sum_{k=0}^{n} \binom{n}{k}=2^n$$ thus is it sufficient to show that (but I think I made a mistake) thus how to do it properly? ...
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63 views

$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$

I want to examine the convergence of the series $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$$ In case it converges I want to evaluate it. I tried the D' Alembert theorem but it was ...
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58 views

Simplify the sum

Consider the sum of two polynomials $$ \sum_{k=0}^{n-1} {{{n-1} \choose {k}}^2 z^{2k}}+\sum_{k=0}^{n-2} {{n-2} \choose {k}} {{n} \choose {k+1}} z^{2k+1}=\sum_{i=0}^{2n-1}a_i z^i. $$ I want to find the ...
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1answer
40 views

Issue to identify binomial distribution

I'm trying to find a general formula for the following case : A production process has one of three possible states: (1) in control, (2) out of control with a type 1 problem, or (3) out of control ...
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1answer
46 views

How to compute $\displaystyle\sum_{k\equiv 1\!\!\pmod{\!4}}\!\!\binom{2014}{k}$?

I have to compute: $S=\binom{2014}{1}+\binom{2014}{5}+\binom{2014}{9}+...+\binom{2014}{2009}+\binom{2014}{2013}$ Could someone help me ?
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0answers
24 views

How to compute $\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$

How to compute the following integral? $$\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$$ in which, $a$ is a positive real number, $n$ is positive integer and ...
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4answers
124 views

How to expand $(x+y)^{-n}$?

How to expand $(x+y)^{-n}$ by binomial theorem, where $n$ is a positive integer? Is there any limitation for $x$ and $y$? If it can be expanded, how to compute the coefficients? Many thanks in ...