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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2answers
67 views

Binomial dependent on a Poisson

I have been working on a problem with a binomial rv dependent on a poisson rv and have worked through to this point: $P(X=x) = \sum_{n=x}^{\infty} \dfrac{n!}{x!(n-x)!} p^x(1−p)^{n−x} ...
3
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1answer
76 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
7
votes
6answers
253 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
3
votes
1answer
59 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
3
votes
2answers
91 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
2
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0answers
57 views

Sum and binomials

I have this sum ...
6
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2answers
190 views

Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $

I stumbled upon the identity $$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$ The right-hand side is a polynomial. ...
1
vote
2answers
86 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
5
votes
2answers
180 views

Binomial expansion inequality

In a paper I am reading, there is a step that seems to come from the following inequality: $$(1+x)^\alpha \le 1+2^\alpha x,$$ where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the ...
5
votes
1answer
136 views

Combinations mod $n$ property

So after some "fooling around" I came across this property in Pascal's triangle (which seems to repeat, and makes a lot of sense): $\begin{pmatrix} n \\ k \end{pmatrix} \mod n = \begin{cases} n ...
1
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0answers
42 views

A Vandermonde like identity for binomial coefficients

The Vandermonde identity is given by $ \left(\begin{matrix} m + n \\ j \end{matrix}\right) = \displaystyle\sum_{j=0}^k \left(\begin{matrix} m \\ j \end{matrix}\right)\left(\begin{matrix} n \\ k-j ...
0
votes
1answer
22 views

Sum of binomial distributed random variables

Let $X \sim Bi(n,p), Y \sim Bi(m,p)$. “Visual arguments” suggest that $X+Y \sim Bi(m+n,p)$. However, I am unable to prove that. Using the definition I can reduce the problem to $$\sum_{i=0}^k ...
10
votes
3answers
234 views

How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$

How do I find sums like these?-- $$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$ that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
1
vote
1answer
151 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
1
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0answers
49 views

Sum involving binomial coefficients

Exist a closed form for $$\left(-1\right)^{N}\underset{i=1}{\overset{N}{\sum}}\left(-1\right)^{i}\dbinom{N}{i}\dbinom{N+i}{i-1}\,\frac{1}{2i+1}?$$ I think I've to use in some way the formula of the ...
4
votes
0answers
176 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
4
votes
0answers
148 views

Proving an equation involving binomial coefficients

Prove that $$\sum_{q=0}^v \binom{v}{q}\frac{q!}{v^{q+1}} = \sum_{q=0}^{v-1} \binom{v-1}{q} \frac{(q+2)!}{v^{q+2}}$$ Thanks. Below are what I have tried: Approach 1: $$\sum_{q=0}^{v-1} ...
2
votes
1answer
51 views

Identity with binomials [duplicate]

Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.
0
votes
1answer
49 views

Binomial distribution or probability intersection

I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = ...
3
votes
2answers
42 views

The asymptotic behavior of the CDF of Binomial distribution

I got stuck with the following problem which seemed not to be very complicated at the beginning! I would like to compute the CDF of a Binomial distribution, \begin{equation*} F(\ell;n,q) = ...
2
votes
0answers
99 views

Sum of product of binomial coefficients and exponential function

I would like to know how to obtain (if it exists) a closed form expression of the sum $$S=\sum^{n}_{k=0}2^k{{n+1}\choose k}{{r-n-2}\choose {n-k}}$$ So far, I have tried to use the method of ...
3
votes
0answers
86 views

Combinatorical interpretation of $\binom{15}{5} = \binom{14}{6}$

I was reading up on Singmaster's conjecture on repeated binomial coefficiencts and I read that $$\binom{15}{5} = \binom{14}{6}$$ Sure, it's possible to prove it non-combinatorically: ...
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votes
1answer
46 views

Dividing Binomial Coefficient

I have a problem which requires multiplying: $$ \frac 1 4 \cdot \binom n k $$ Expanded: $$ \frac 1 4 \cdot \frac {(n)!} {5!(n-5)!} $$ The answer is below, but it isn't clear how to get from the ...
0
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1answer
26 views

Finding $V(X)$ when you don't have a density/distribution function.

I just did the first part of this problem: You have a lot of $50$ items and are taking a sample size of $15$. In the lot $3$ items are defective. The lot is accepted if the number of defective items, ...
2
votes
2answers
87 views

Alternating sum of a simple product of binomial coefficients

I would like to evaluate the following alternating sum of products of binomial coefficients: $$\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$$ I had the idea to use Pascal recursion to re-express ...
0
votes
2answers
73 views

Show that $\binom{n}{k}< \binom{n}{k+1}$ if and only if $k < (n-1)/2$ [closed]

Show that $\binom{n}{k} < \binom{n}{k+1}$ if and only if $k < \frac{n-1}{2}$ and then use this to deduce that the maximum of $\binom{n}{k}$ for $k=0,1,\dots,n$ is $\binom{n}{\lfloor ...
0
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0answers
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 ...
9
votes
3answers
245 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...
1
vote
3answers
47 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
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2answers
69 views

Prove Maximum term in the expansion.

How should I prove the maximum term in the expansion of $(x+a)^n$ where $ax>0$ is the term $C(n,r)x^{(n-r)}a^r$ for which $r= \left[\cfrac{(n+1)}{(n/a)+1} \right]$ ?
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1answer
33 views

Proving an identity involving binomial coefficients and fractions

I've been trying to prove the following formula (for $n > 1$ natural, $a, b$ non-zero reals), but I don't know where to start. $$\sum_{j=1}^n \binom{n-1}{j-1} \left( \frac{a-j+1}{b-n+1} \right) ...
1
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1answer
28 views

Binomial thereom to figure out coefficents

Use the binomial theorem to find the coefficient of $x^8y^5$ in $(x + y)^{15}$ My textbook shows how to do this looking at the coefficents of Pascal's triangle but, I know theres another way using ...
0
votes
1answer
32 views

Equation with binomial coefficients

Problem: Find the roots of $6z^5+15z^4+20z^3+15z^2+6z+1 = 0$. What I found: I realized that the coefficients were the binomial coefficients of $6$. Putting these values in, you would get ...
5
votes
1answer
104 views

Binomial Congruence

How can we show that $\dbinom{pm}{pn}\equiv\dbinom{m}{n}\pmod {p^3}$ for positive integers m and n and p a prime greater than 5? I can do it for mod p^2 but Im stuck here.
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6answers
55 views

Distribution of a binomial variable squared

If I know $X$ is a binomial random variable, how can I find the distribution of $X$ squared (I know that $P(Y=y=x^2) = p(X=x)$ but does this distribution have a standard name)? In particular, how can ...
5
votes
1answer
99 views

Proof of Interesting Binomial Identity

In my work I've come across the interesting binomial identity $$ \sum_{n\geq k} \frac{\binom{n}{k}}{\binom{m-1}{k}} \frac{\binom{m-1}{n} \binom{i-m-1}{j-n-1}}{\binom{i-2}{j-1}} = ...
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votes
1answer
71 views

Simplifying the sum of two consecutive binomial coefficients

I've been trying to figure out how they simplified the right side of the equation all the way down. $$\eqalign{ \binom nk + \binom n{k-1} & = \dfrac{n!}{(k)!(n-k)!}+\dfrac{n!}{(k-1)!(n-k+1)!} \\ ...
0
votes
1answer
68 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
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1answer
52 views

Binomial Coefficient: monotonically decreasing in this range?

relating to this question, I'd like to ask a further one. Again we have $$f(x)={k-1 \choose x-1} p^x (1-p)^{k-x}$$ We know that this term is maximal for $x=kp$, before increasing, afterwards ...
0
votes
1answer
26 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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0answers
56 views

Is there a sharper bound than exponential for $\sum_{k\ge0}\frac{m!(k+n-m)!}{(k+n)!}\frac{s^k}{k!}$?

I am trying find a bound for an expression and I am getting something not quite as convenient as I hoped. Going through my calculations again I think that the only place I use a non sharp bound is ...
3
votes
2answers
79 views

Integer sum as binomial coefficient

What's the rule for expressing integer sums as binomial coefficients? That is, for $p=1$ it is $$\sum_{n=1}^N n^p = {{N+1}\choose 2} $$ What is it for higher powers?
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2answers
35 views

Alternative proofs:

I have worked about a result and want to know if there are better ways of proving the following: $$N^M - (N-1)^M$$ $$=\binom{N}{1}(N-1)^{M-1} + \binom{N}{2} (N-1)^{M-2} + \binom{N}{3} (N-1)^{M-3} + ...
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0answers
27 views

Equation for those level curves?

Related to this question Let $n\in\mathbb{N}^*,\alpha\in\mathbb{R}$. What would be the equation $y=f(x)$ for the curve defined by $\ln\binom{N-y}{x}=\alpha$ That's how they look : TL;DR : What ...
0
votes
1answer
67 views

The binomial coefficients $\binom{n}{ p}$ are divisible by a prime $p$ only if $n$ is a power of $p.$

I'm looking for a "high school / undergraduate" demonstration for the: All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ ...
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1answer
34 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
0
votes
0answers
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 ...
2
votes
0answers
53 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$$ which can be proved combinatorically whether one particular element(among the $n$) is ...
0
votes
1answer
30 views

A little help in this binomial problem.

In the binomial expansion of $(a-b)^n$, $n \geq 5$, the sum of $5^{th}$ and $6^{th}$ term is $0$, then $\frac{a}{b}$ is ? I've solved this problem but its coming $n-4$ only, and the answer says it ...
0
votes
3answers
43 views

Binomial coefficient after expansion.

What is the coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3)^6$ ?