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 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
24 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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38 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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0answers
67 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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17 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
46 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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123 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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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
16 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) ...
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1answer
25 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
23 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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77 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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53 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 ...
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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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34 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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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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22 views

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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59 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
56 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
22 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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${n \choose r}=\frac {n!}{r!(n-r)!}$ 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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25 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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38 views

Binomial coefficient after expansion.

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

A generalized combinatorial identity for a sum of products of binomial coefficients

I have the following question. For given natural numbers $n$ and $d$, let $a_1,a_2,..., a_r$ be fixed integers such that $a_1+\cdots+a_r=d$. Let $A=\{(i_1,..,i_r)~|~0\le i_j\le n~ \text{and}~ ...
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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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24 views

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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109 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
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30 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
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28 views

Number of draws that contain at least $k$ red coins

Assume there are $n$ coins in an urn from which $r$ are read. What is the number of draws of $r$ coins that contain at least $k$ red coins? It is obvious that there are ...
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57 views

Limit of binomial coefficients

Let $0\leq a_n\leq n$ be a sequence of integers. Under which condition on the $a_n$ does $$\frac{{n-a_n\choose a_n}}{{n\choose a_n}}=\frac{(n-a_n)(n-a_n-1)\dots(n-2a_n+1)}{n(n-1)\dots(n-a_n+1)}$$ ...
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highest power of prime $p$ dividing $\binom{m+n}{n}$

How to prove the theorem stated here. Theorem. (Kummer, 1854) The highest power of $p$ that divides the binomial coefficient $\binom{m+n}{n}$ is equal to the number of "carries" when adding $m$ ...
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Coefficient of polynomials

Could someone explain to me why $$ [x^{24}](1-2x^6)^{-31} = 2^4 \binom{4 + 31 - 1}{31 - 1} \, ? $$ Reads: The coefficient of $x^{24}$ in $(1-2x^6)^{-31} =$ ...
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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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1answer
32 views

How prove $\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$

we know Bernoulli number such identity $$\sum_{k=0}^{n}\binom{n+1}{k}B_{k}=0$$ see:Bernoulli number identity show that $$\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$$ where $B_{n}$ ...
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184 views

Generating function of binomial coefficients ${n\choose5}$

How to prove easily this identity for (almost classical) series with binomial coefficients: $$ \sum_{n=5}^\infty \dfrac{\binom{n}{5}}{2^{n+1}} = 1 . $$ Thank you. Any smart proof would be much ...
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71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
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1answer
24 views

Equality of binomial coefficients

I have seen that the following equations are equal, but are wondering how this is shown ${n \choose m} \cdot 1 \cdot 3 \cdots (2m-1)\cdot 1 \cdot 3 \cdots (2n-2m-1) = \frac{n!}{2^n} {2m \choose m} ...
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61 views

Extracting a coefficient from a generating function

Background: I am working on an exercise relating to Skolem $k-$subsets with index $k$ in Goulden and Jackson's Combinatorial Enumeration text and they broke it down to finding the coefficient of $x^n$ ...
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2answers
25 views

Probability of special configuration of ones in a binary string

Consider the sequence $(X_i)_{1 \leq i \leq L}$ of i.i.d. random variables, where $X_1 \in \{0,1\}$ and $P(X_1 =1) = p$. For a $k \in \mathbb{N}$ define the event $A_{k,L}$ as "all ones in the ...
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2answers
55 views

Largest term in binomial expansion of $(1+n)^k$

The binomial expansion of $(1+n)^k$ is $$(1+n)^k=1+\binom{k}{1}n+\binom{k}{2}n^2+\cdots+\binom{k}{k}n^k.$$ If $n=1$, then the term in the middle is the largest, i.e. when $i=\lfloor k/2\rfloor$ and ...
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1answer
23 views

How find this sum $\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$

Let $m$ is give postive integer numbers, Find the sum $$\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$$ if this not have closed form,and can you use Special function ?