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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What is the alternating sum of coefficients and what does it have to do with the zeroes of the function?

So, my teacher told us today, while we were solving this integral: $$\int\frac{dx}{x(2x^3+x^2+1)}$$ that the alternating sum of coefficients of $2x^3+x^2+1$ is 0 (2-1+0-1=0) and hence, one zero of the ...
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86 views

What is the sum of this infinite series

So the question is - $\displaystyle S = \sum_{n=1}^\infty{\frac{1}{10^n}\left(\begin{matrix}2n\\ n\end{matrix}\right)}$. Find $S$. I tried converting the $n^{th}$ term as a difference of two terms ...
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4answers
71 views

Equality of sums

How does one show that $$\sum_{j=k}^n\binom{j-1}{k-1}q^{j-k}=\sum_{j=k}^n\binom{n}{j}p^{j-k}q^{n-j},$$ where $p+q=1$? I suppose one needs to substitute $p=1-q$ on the right side and then use the ...
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4answers
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What is the story behind ${n+1 \choose k} = {n \choose k} + {n \choose k-1}$? [duplicate]

By exploring the inductive proof from this question I came to the point where I did not understand this step: $${n+1 \choose k} = {n \choose k} + {n \choose k-1}$$ There is a wikipedia article but ...
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1answer
59 views

Discrete Math Identity Proof Binomial Coefficients

The question is to prove this identity: ! where k, m, n ∈ Z+. Using pascal's identity on the left, so far I have: ! If m is even then they cancel each other and should equal 0. If m is odd then ...
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1answer
47 views

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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1answer
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
39 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
62 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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1answer
52 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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2answers
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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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55 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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1answer
42 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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1answer
19 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
53 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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515 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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1answer
220 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
39 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
24 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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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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37 views

Binomial Theorem Help

In my assignment I have the following question: Alan
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1answer
63 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
101 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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30 views

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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1answer
39 views

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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2answers
107 views

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
32 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
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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
67 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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1answer
27 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
52 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
26 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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4answers
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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
33 views

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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2answers
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
21 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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2answers
120 views

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
224 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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4answers
48 views

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
43 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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1answer
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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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2answers
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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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0answers
31 views

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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4answers
64 views

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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2answers
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 ...