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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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 & = & ...
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17 views

Bound the expression with Stirling approx.

I already compute 2 expressions of one problem, but they are so complicated as below. My prof requests me to reduce them or upper bound them by Stirling approximation but I have not succeeded. 1/ ...
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2answers
81 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 ...
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40 views

Sum and binomials

I have this sum ...
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1answer
136 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. ...
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2answers
70 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?
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41 views

An equation on Catalan number [closed]

Catalan numbers have the form $C_n=\frac{1}{n+1}\binom{2n}{n}$ prove: $C_{n+1}=\sum_{m+k=n}C_mC_k$ I tried to expand $C_n$ but soon get confused..
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2answers
73 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 ...
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0answers
89 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 ...
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33 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 ...
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1answer
15 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 ...
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3answers
107 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..?
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1answer
129 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 ...
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22 views

relative error of Poisson approximation to sum of Binomial

We have given $X_i\sim Bin(n_i,p_i)$ for $i \in \{1,...,m\}$ and are interested in $$P[X \geq x]$$ for $X=\sum_{i} X_i$. As we can approximate $X_i$ by $Y_i \sim Poisson(n_i p_i)$, I wonder, ...
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34 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 ...
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1answer
124 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 ...
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124 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} ...
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1answer
46 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.
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1answer
27 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) = ...
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2answers
25 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) = ...
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46 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 ...
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71 views

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

I was reading up on Sigmaster'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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1answer
45 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 ...
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1answer
23 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, ...
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2answers
50 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 ...
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2answers
67 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 ...
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21 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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127 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 ...
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3answers
29 views

Greatest value of the binomial coefficient.

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
26 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
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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) ...
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1answer
26 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 ...
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1answer
26 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 ...
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1answer
78 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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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 ...
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1answer
79 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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1answer
44 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
37 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 ...
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1answer
23 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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Denominator of rational binomial coefficients

I am having trouble proving that $\displaystyle\binom{k}{1/2}$ has a power of 2 in its denominator, $\displaystyle\binom{k}{1/3}$ has powers of 3 in the denominator. I think inductive approach ...
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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 ...
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2answers
65 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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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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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 ...
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1answer
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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
23 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 ...
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49 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

${n \choose r}=\frac {n!}{r!(n-r)!}$ 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 ...
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1answer
26 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 ...
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3answers
39 views

Binomial coefficient after expansion.

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