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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How to find $\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$?

If n is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
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Computing a sum involving binomial coefficients

I am doing some (pretty heavy) computations, and I am stuck at a point that can be rephrased as follows: Let $m>n\ge0$ be two integers. Compute ...
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28 views

Binomial Theorem coefficient sum…

Recently I encountered a question but its answer as well as the way the author of the book has solved the question seemed wrong to me.. Find the sum of the coefficients of the expansion of ...
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Number of terms in multivariate polynomial

We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have: ...
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40 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
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identity on Pascal's triangle modulo 2

Consider Pascal's triangle with entries modulo $2$, and let $(k,l)$ denote the $l$-th entry in the $k$-th row by $(k,l)$. Show that, for all $n \in \mathbb{N}$, each entry of the triangle with ...
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Best way to expand $(2+x-x^2)^6$

I've completed part $(a)$ and gotten: $64+192y+240y^2+160y^3+...$ Using intuition I substituted $x-x^2$ for $y$ and started listing the values for : $y, y^2 $ and $y^3,$ in terms of $x$. ...
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Find $k$ if given the constant term of a binomial expression?

Consider the expansion of $x^2(3x^2+\frac{k}{x})^8$. The constant term is $16,128$. Find $k$. This is simply an example of a type of question I cannot understand how to do. I have many questions: 1) ...
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continued fraction $F(x)$ that is a generating function of central binomial coefficients

Given the following continued fraction $$F(x) =\cfrac{1}{x+\cfrac{2^2(2^2-1)}{6x+\cfrac{3^2(3^2-1)}{12x+\cfrac{4^2(4^2-1)}{20x+\cfrac{5^2(5^2-1)}{30x+\ddots}}}}}=\frac{1}{\sqrt{x^2+4}}$$ Then ...
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Anti diagonal elements of table forming pascal traingle

A function in $k$ and $n$ leads to the formation of this table. The elements in this table are rows of pascal triangle if we look at the anti diagonals elements of this table. They have also been ...
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$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$

2$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=2\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$. This is an identity in a note for a class in Markov Processes, but I can't ...
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878 views

A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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42 views

Problem based on sum of binomial coefficients

Let $m$ be the smallest positive integer such that Coefficients of $x^2$ in the expansion $\displaystyle (1+x)^2+(1+x)^3+.....+(1+x)^{49}+(1+mx)^{50}$ is $\displaystyle (3n+1)\binom{51}{3}$ ...
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63 views

Combinatorial proof of a certain alternating sum of binomial coefficients

The following identity appeared as a question earlier today $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 ...
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whether is $\sum_{k=0}^{j-1} \binom{i}{k}=\sum_{k=0}^{j-2} \binom{i-1}{k}+\sum_{k=0}^{j-1} \binom{i-1}{k}$ true or false?

I have tested some trivial samples when $j = 1,2,3$. But I can't prove if it is true or false generally. Any help would be great, thanks!
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Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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31 views

Vandermonde-type convolution with geometric term

Is there a closed-form solution to the following sum? \begin{align*} f(r, s, n) = \sum_{k=0}^{n}c^k\binom{r}{k}\binom{s}{n-k} \end{align*} I know this corresponds to find the coefficient of $x^n$ of ...
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Seating children in the cinema

I just had finished my class and have been struggling with a problem. There's $9$ seats in the cinema, and two families $F_a=\{F_1,F_2,F_3,F_4,F_5\},$ $F_b=\{F_a,F_b,F_c,F_d\}$ In how many ways can ...
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2answers
72 views

Odd binomial sum equality has only trivial solution?

Suppose $$\sum_{k\ {\rm odd}}^n {n \choose k} 2^{(k-1)/2} = \sum_{k\ {\rm odd}}^m {m \choose k} 2^{(k-1)/2} 3^{(m-k)/2}.$$ Does $m=n=1$? Clearly $m \leq n$, and for every $n$ there is at most one ...
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Please help me compute this$ \sum_m\binom{n}{m}\sum_k\frac{\binom{a+bk}{m}\binom{k-n-1}{k}}{a+bk+1}$

Compute following: $$ \sum_m\binom{n}{m}\sum_k\frac{\binom{a+bk}{m}\binom{k-n-1}{k}}{a+bk+1} $$ Only consider real numbers a, b such that the denominators are never 0. Now I simplify it into $$ ...
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Evaluate the combination of $\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$

Can any one help me please to get the approximate result of this combination problem using asymptotic notation: $$ \sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j} $$ Thanks
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T0 show an equation by using binomial theorem

$$\left(1+\frac{a}{n}\right)^{(n-k)} = e^a \left(1-\frac{a(a+k)}{2n}\right)+o\left(\frac{1}{n}\right)$$ as $n\to\infty$. How the binomial theorem show this above equality? Thank you for your help!
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72 views

Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$

I need to proof this following equality : $$\binom{m+n}{m}=\sum_{i=0}^m \left(\binom{m}{i}\binom{n}{i}\right)$$ This is what I did combinatoric proof: Left : subset with $m$ members from $m+n$ ...
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Coefficient of product of polynomials.

Suppose we have the polynomial $f(x)$ and another polynomial $g(x)$. How can I find the coefficient of say $x^n$ in the product of the polynomials without actually multiplying. I am not that ...
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Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
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28 views

How to choose $n$ balls from the bags?

Given $4$ bags A, B, C and D. Bag A contains 'a' number of balls. Bag B contains 'b' number of balls. Bag C contains 'c' number of balls. Bag D contains 'd' number of balls. I have another bag E ...
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Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
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29 views

Newton Binomial Problems

Let there be this binomial: $$ (\sqrt{2} + \sqrt[3]{3})^{8}$$ How many rational terms are there in it's development? I tought that the number of terms is given by n + 1 = 8 + 1 = 9, but that doesn't ...
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Can anyone verify $\int_{0}^{\infty}\frac{e^{-2nx}+2nx-1}{x(e^x+1)}dx=\ln{2n\choose n}$? [closed]

Central binomial coefficient from mathworld $$\frac{2^{2n+1}}{\pi}\int_{0}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx={2n\choose n}$$ Here we have $\ln{2n\choose n}$ in term of another integral, ...
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Coefficient of $x^n$ in binomial expansion

I want to find the coefficient of $x^n$ in $G(x)$ where $ G(x) = \frac{1}{1-x^{a_1}}\times\frac{1}{1-x^{a_2}}\times\dots\times\frac{1}{1-x^{a_k}}$ how do I approach this? It would be helpful if it ...
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34 views

Experiment by Bernoulli process

I have a question. Assume I carry out an experiment by Bernoulli process. I repeat the tests until the number of successful outcomes exceed the number of unsuccessful outcomes by m. What will be the ...
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51 views

Intuitive explanation of $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$

Could anyone please explain me the reasoning behind this formula? $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$ Thanks so much!
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What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$.

What is $\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n}$ for $x > 4$. Here is what I got so far (using Cauchy's integral formula) : $$\sum\limits_{n=0}^\infty \binom{2n}{n}\frac{1}{x^n} ...
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1answer
42 views

Proving binomial identities [duplicate]

Can someone help me prove these two binomial identities using either walks in Pascal's triangle or a committee-selection model? $(1)$ $\qquad$ $\displaystyle\sum_{k=0}^m {m\choose k}{n\choose ...
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40 views

Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I ...
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Probability of winning a prize in a raffle (that each person can only win once)

There is a raffle coming up. 4000 tickets have been sold, and there are 10 prizes to win. I have bought 8 tickets. What are the odds I will win a prize? Note: each person can only win once. There ...
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44 views

Binomial coefficient of power n

How can I find the coefficient of $x^n$ using binomial theorem? $$\frac{1-x}{(1+x)^3}$$
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Self-avoiding walks from one diagonal to the other on $mxn$ lattice is ${m+n \choose m,n} $

According to wikipedia "self-avoiding walks from one end of a diagonal to the other, with only moves in the positive direction, there are exactly $$ \binom{n+m}{n,m} $$paths for an $m × n$ ...
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How can I find the cubic polynomial, using $4\times4$ linear system with coefficients? Or any help of reference?

part A in standard form $$y = a_0 + a_1t + a_2t^2 + a_3t^3$$ passes points $$(0, 4), (1, 3), (−1, 7), (2, −2)$$ part B and for the same cubic polynomial in shifted basis $$\{1, t − 2,(t − ...
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How many $5$ card poker hands contain at least $1$ red and $1$ black card?

How many $5$ card poker hands contain at least $1$ red and $1$ black card? I used inclusion-exclusion to calculate my answer. The number of total poker card hands are:$$52\choose 5$$I have $26$ red ...
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40 views

Extending binomial identity $ \sum\limits_{k=0}^n\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$ to $0<x<1$

I found in Matlab that $$ \sum_{k=0}^n~\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$$ for $1\leq x< n$ only (I am about 95% sure of this since the sum is numerically unstable and cannot give ...
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Prove $\left(\dbinom nk \right)= \left(\dbinom{k+1}{n-1}\right)$ [closed]

I need to prove $\left(\!\dbinom nk \!\right)= \left(\!\dbinom{k+1}{n-1}\!\right)$ where the double parens denote multiset coefficients and $n,k$ are integers with $1 ≤ k≤ n$ using an algebraic proof. ...
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28 views

Proof binomial coefficient [closed]

I'm trying to prove the following: $$\binom{n + p}{k} = \sum_{j=0}^n \binom{n}{j} \cdot \binom{p}{k - j}$$ How do I do it? Induction? And can someone hint me at how to start?
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61 views

Easy geometric sum with binomial coefficient

In the context of stochastic processes I came across the following equality, where $|s| < 1, p \in [0,1]$: $$\sum^\infty_{k=0}(s^2p(1-p))^k\begin{pmatrix} 2k \\ k \end{pmatrix} = ...
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61 views

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$. I know he Binomial Theorem will be useful here, but I am struggling to use it with any certainty.
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1answer
20 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
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1answer
31 views

Using the Binomial Identity, prove that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Using the Binomial Identity, prove that: $${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$$Because this is in the form of a Binomial Coefficient, I can break down the LHS ...
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4answers
28 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus ...
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1answer
28 views

Show that $2k\choose k$ divides the lcm of $1, \dots, 2k+1$

I want to show that $(2k+1){2k\choose k}$ is a factor of $\text{lcm}(1, \dots, 2k+1)$. Clearly the divisor is equal to $2^k\frac{1\cdot3\cdot\dots\cdot (2k+1)}{k!}$, but I don't know how to show that ...
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1answer
19 views

calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$

Is there any formula for calculating $\sum_{l=0}^{\infty}\binom{l+100}{l}0.5^l 0.5^{100}$ and $\sum_{l=0}^{\infty}l \binom{l+100}{l}0.5^l 0.5^{100}$?