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 prove that ${l \choose a_1,…,a_n}\le n^{l-1} $ , when $a_1+…+a_n=l$.

In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then $${l ...
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568 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
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Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even

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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Proof of Pascal's rule by induction [duplicate]

I'm trying to prove pascal's rule by induction.Who could tell me how can I prove the following equation. i.e $$\dbinom{n}{k}=\dbinom{n-1}{k-1}+\dbinom{n-1}{k}$$ Many Thanks!!!
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Proving Pascal's identity

So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$ I know you can use basic algebra or even an inductive ...
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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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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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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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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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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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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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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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This question involves Pascal's Triangle & Binomial Theorem. Full Question

Could anyone please help me on the following problem: Factorize the expression $P(n)=n^x-n$ for $x=2,3,4,5$ Determine if the expression is always divisible by the corresponding $x$. If divisible use ...
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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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881 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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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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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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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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binomial calculation method

I want solve this probability: For $p= 0.4$ $q=0.8$ $n= 20$ $1-P(5<x<11)$ = $1-\sum_{k=6}^{10} \binom{20}{k}(0.4)^k(0.6)^{20-k}. Wolfram Alpha -> = 0,2531$ Is calculation method ...
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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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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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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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Inverse Euler theorem to calculate $\binom{n}{r}$

How can we use Inverse Euler theorem or properties to calculate the binominal coefficients or say $\binom{n}{r}$? What is the algorithm for this ? An example for the same will be greatly ...
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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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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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Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Consider the following identity: \begin{equation} \sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1} \end{equation} Consider the set $S$ of size $2n-2$. We partition $S$ into two sets $A$ ...
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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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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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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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2answers
123 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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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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642 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
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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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1answer
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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2answers
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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Formula for a variation of Pascal´s Triangle

So recently I came across a math problem, which states the following: You play a game with a fair coin. You start with zero points, and you throw the coin. If you throw heads, you add 1 point to the ...
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30 views

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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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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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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1answer
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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3answers
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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2answers
55 views

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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2answers
40 views

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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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 $m × n$ ...