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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Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
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242 views

proving inequality for combinatorial sum

If somone can prove the following for every $d\leq r$ (for $d=0,1$ its easy, see below, the case d=r may be also simple, I didn't find something helpful) $$\frac{(d!)^2}{2^{n-2d}}\sum_{k=0}^{n}{n ...
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141 views

Compute the value of a sum (as an expression involving one or two binomial coefficients)

I've been asked to compute the value of a sum. The answer should be an expression involving one or two binomial coefficients. The initial expression: $$ \sum_{k} \binom{80}{k} \binom{k+1}{31} $$ ...
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46 views

is this binomial distribution correct?

I am trying to work out what is the chances of getting the following marks $(100\%, 70\%, 60\%, 50\%)$ in a paper containing $50$ questions, each question containing yes/no options. Using Binomial ...
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178 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
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407 views

Binomial coefficient modulo prime power without generalized Lucas theorem

I've been working on this problem computing $n \choose r$ for large $n$ and $r$, modulo a composite. I could implement the generalized lucas theorem to handle the prime power case, but I want to ...
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144 views

Binomial coefficient (mod m)

$n\ge k $ $ m>0$ How to find? $$\binom{n}{k}\mod m$$ without counting $n,k!$
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Recognizing binomial mixtures

I'd like to know a procedure to recognize whether a given probability distribution over outcomes $\{0, \dots, n\}$ can be expressed as a mixture of $n$-trial binomial distributions with different ...
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31 views

For $X \sim \mathrm{Binomial}(n,\frac{1}{2})$ does there exist $a,b,c,Y$ s.t. $\Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$?

I need to upper bound some complicated expressions involving binomial distributions: Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$. I want to find $a,b,c,m$ such that for $Y \sim ...
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108 views

Binomial coefficient intervals (inequality)

For given $N$, $x$ and $k$ such that $0\leq x<N$ and $2\leq k\leq \left\lfloor \frac{N+1-2x}{2}\right\rfloor $, does it exist $p,$ $2\leq p\leq \left\lfloor \frac{N+1}{2}\right\rfloor $ such that ...
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270 views

The Lucas Theorem and facts

I have studied the Lucas theorem and I encountered the following facts. How to deduce the following facts from The Lucas theorem? (1) If d, q > 1 are integers such that , $$\binom{nd}{md}$$ ...
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181 views

Lucas' theorem Consequence

Lucas' theorem consequence $$\binom {m}{n}=\ \prod_{i=0}^k\;\ \binom {m_i}{n_i} \pmod{p},$$ $$m=m_k\;p^k+m_{k-1}\;p^{k-1}+\cdots +m_1\; p+m_0,$$ $$n=n_k\;p^k+n_{k-1}\;p^{k-1}+\cdots +n_1\;p+n_0$$ ...
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63 views

calculation of the sum using idea of one answer

I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ? ...
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290 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
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49 views

Inexpressibility of certain coefficients and discrete versions of Hölder's theorem

In an answer to a recent question, I noted that there were probably no explicit formulas for Stirling numbers (of the first kind, specifically) and speculated that this might be coupled to a sort of ...
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143 views

n-th self discrete convolution

Lets define discrete $ f_N(i) = 1,\space i = 1...N $ I need to find $ G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m} $ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
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191 views

Interdependent constraints combination problem

I am trying to solve the following combination problem. You have 4 knobs or levers that have maximum values, such as 0-20, 0-30, 0-50 and 0-100. Their total values must equal an amount, say 47. Their ...
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Proving that $\Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. You are given that $\displaystyle L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}]=\sum_{k=0}^{j} ...
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24 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
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32 views

Closed Form for a Sequence

I have come across this sequence $$a_0 = -2, a_1 = 5, a_2 = -28, a_3 = 255$$ and, in general $$a_n = -\frac{1}{2}\bigg(\sum_{i=1}^n \binom{2n+4}{2i}a_{n-i} + \binom{2n+4}{2n+1}\bigg)$$ I've tried ...
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Sampling combinations (from a binomial coefficient) without replacement

The total number of combinations of $k$ items out of $n$ total is $n \choose k$, or a binomial coefficient. This can be a very large number even for pretty small $n$. The binomial coefficient ...
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24 views

n-th degree Bezier curve-bernstein polynomial

I want to make a code that will draw n-th degree Bezier curve which would be calculated through Bernstein polynomials.My problem is not related to code writing,but its math kind.Reason I have ...
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25 views

Sum of binomial coefficients multiplied

How to approximate the next one relative to $m$ $$ \sum_{k=0}^m \binom {n}{n-k} (n-k-\frac k{\sqrt{2}})^2? $$ Or for example the simplier sum $$ \sum_{k=0}^m \binom {n}{n-k} (n-k)^2? $$
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Relation between Hyperfactorial, Superfactorial, Pascal's Triangle and Binomial Coefficient

I read here that the product of the elements in the $N^{th}$ row of Pascal's triangle is equal to $(n!)^{n+1}/(\prod_{k=1}^n k!)^2$. Let's call the product of elements in the $i^{th}$ row of Pascal's ...
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23 views

Can one interchange the limit and summation in this example?

Let $L$ be a positive real number and $a$, $b$ and $x$ real numbers strictly between $0$ and $L$. For integers $m$ and $n$, define $$ A_{m,n} := \sum_{k=1}^{[\frac{\sqrt{n}(b-a)}{2}]} \frac{1}{2^n} ...
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44 views

What is known about this Pascal's Triangle problem?

Suppose for the $k$th row of Pascal's Triangle, you want to take each of the integers $1..k$ and multiply it by a different number in the row, then take the sum. For example, for $k=4$, we have ...
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28 views

Find the largest term for the binomial expansion of (n+1/n)^(2n+1)?

For (∀n ∈ N{1,2 }). So what I have done at first to try the (Tk+1/Tk)>=1, but it gave me wrong answers, for example I got that my coefficient was negative which is absurd.
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Can binomial coefficient be defined as a natural number if n is the cardinality of a countable set?

Can binomial coefficient n choose k, k less than or equal to n, be defined as a natural number if n is the cardinality of a countable set?
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Permutation and combination identity

Prove that $\displaystyle \sum_{i=0}^n \binom{n}{i}\binom{m+i}{n}=\sum_{i=0}^n \binom{n}{i}\binom{m}{i} 2^i$ for natural numbers $m,n.$ The question doesn't seem to have any direct combinatorial ...
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23 views

Choice problem: number of pairs can be formed out of a set with odd cardinality

There are 17 languages (at a meeting) and for every two languages there is one interpreter assigned. The number of pairs we can form out of 17 languages is $\binom{17}{2} = 136$. So 136 interpreters ...
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difference between independent binomial variables

It is well known that if $X \sim B(m, p)$ and $Y \sim B(n, p)$ are independent then $X+Y \sim B(m+n, p)$ but what is the distribution of $X-Y$? Here is what I have tried. $\Pr[X-Y = c] = \sum_{i=0}^n ...
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52 views

How binomial theorem is used in IP distribution?

I have read the binomial theorem/ Pascal triangle that they can be useful for IP ( Internet Protocol) address distribution . However, I am unable to understand it. How it can be applied for IP address ...
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38 views

How to add binomial coefficients?

How do I add let say $\binom{n-2}{k-2} + \binom{n-2}{k-1}$ What are the steps of adding these together without breaking them down in factorials and add them up?
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All or no Heads from biased coin tosses

What is the probability that all five tosses of a biased coin with $P(H)=0.28$ are (a) Heads and (b) Tails? (c) What is the probability of at least one Head? (a) Heads $Pr(5\ Heads) = {5 \choose 5} ...
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Transformation of a sum

I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ ...
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is binomial congruence given in article true or false?

I'm just reading a paper which, on its page 3, Application 8, claims the following: $$\binom{k+sp}{j}\equiv\binom{k}{j}\pmod{p}$$ where $p\ge 1$, $s\ge 1$, $k\ge 1$ and $p\not\mid j$ (actually, it ...
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Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
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Bounding a specific function of binomial coefficients

While trying to directly prove the existence of expander graphs (e.g. http://www.cs.toronto.edu/~avner/teaching/S6-2414/TUT2.pdf), one uses the following inequality: $$\sum_{s=1}^{n/2} ...
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72 views

Generalized Leibniz Rule

Leibniz Rule states that, $$(f\cdot g)^{(m)}(x)=\sum_{k=0}^m \binom{m}{k} f^{(m-k)}(x)g^{(k)}(x).$$ Writing this with differentiation denoted by $D$, we might say $$D^m (fg) = \sum_{k=0}^m ...
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$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
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Lower bound for a relative of the central binomial coeff

The central binomial coefficients $\binom{2m}{m}$ have g.f. $\frac{1}{\sqrt{1-4z}}$ and lower bound $\frac{4^m}{\sqrt{4m}} \le \binom{2m}{m}$. I'm interested in a related integer series $$T(2m, m) = ...
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27 views

Simplification of a power weighted alternating binomial sum

Given positive integers $T$, $n$ and $m$ and real number $p$ with $0< p < 1$, how can I simplify the following binomial sum: $$ \sum_{k=m}^{\lfloor ...
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55 views

Asymptotic complexity of $\sum_{k=1}^m \binom{2^m}{2^k} \binom{2^k}{2^{k-1}}$

I'm trying to examine the asymptotic complexity of $$f(m) = \sum_{k=1}^m \binom{2^m}{2^k} \binom{2^k}{2^{k-1}}$$ Question: How do you prove or disprove $f(m) \in \mathcal{O}(2^{2^m})$? Bonus ...
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42 views

A sum of powers of binomials

For $n$ and $k$ non-negative integers, let $$F(n,k) = \sum_{i=0}^{n}\binom{n}{i}^k.$$ For example, $F(n,0)=n+1$, $F(n,1)=2^n$ and $F(n,2)=\binom{2n}{n}$. Does there exist a general formula for ...
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Finding the co-efficient

I am trying to find the co-efficient of $\frac{1}{z}$ in the expansion of $$\frac{(1+z^2)^{2n}}{z^{2n+1}}$$ I proceeded like this - ...
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64 views

What's the interpretation of $\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$?

I'm having problems with finding the combinatorial interpretation of this sum: $$\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$$ Can anyone help, please?
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Confidence intervals for bernoulli trials over a cyclic time series

I have a time series with observations of 0 or 1 observed yearly for approx. 20 years. The time series is cyclic and I want to find a CI for the probability p over the cycle (mean). Unfortunately I ...
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59 views

$\sum_{k=1}^{n}\dfrac{(-1)^{k+1}}{k}{n\choose k}=\sum_{k=1}^n\dfrac{1}{k}$

If $n$ is a positive integer, then the above identity holds. I tried to solve this question using generating function. $$A(x)=\sum_n\left(\sum_{k=1}^n\dfrac{1}{k}\right)x^n=-\dfrac{\log(1-x)}{1-x}$$ ...
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31 views

Efficient way to compute the binomial using $(2^k+1)^{k+1}$

The following web page: "http://introcs.cs.princeton.edu/java/78crypto/" (at Exercise 28) effectively says that: "Pascal's triangle. One way to compute the $n$-th row of Pascal's triangle (for $n ...
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40 views

help with a double summation with binomial coefficients

I was stuck in showing the following derivation step in a book. $\sum \limits_{c=1}^{d} \sum \limits_{j=1}^{c} \binom{c}{j} \binom{d}{c}(-1)^{d-c}\delta_{j}^{2} = \sum \limits_{j=1}^{d} \{ \sum ...