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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Proof that repeated sum equals binomial formula

Let $s, d$ be positive integers. Can you prove the following general formula for the repeated sum? I developed this problem on my own, but is it a well known result? $$\sum_{i_1 = 0}^s \sum_{i_2 = ...
2
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2answers
49 views

Nested sum $\sum_{i<j< \cdots < k} ij \cdots k$

I am wondering if there is any known closed form for the following nested sum? : $$ \sum_{i<j<\cdots <k} ij\cdots k $$ where each $i,j,\cdots,k =1, \cdots, n$ I tried the first one: $$ ...
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3answers
34 views

Calculate sum $S=\sum_{k=0}^{n}\begin{pmatrix} k \\ m\end{pmatrix}$

Calculate sum $$S=\sum_{k=0}^{n}\begin{pmatrix} k \\ m\end{pmatrix}$$ My solution if $n<m$, $S=0$ else $$S=\sum_{k=0}^{n}\begin{pmatrix} k \\ m\end{pmatrix}=\sum_{k=m}^{n}\begin{pmatrix} k \\ ...
2
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3answers
59 views

The value of the sum $\binom {20}0 -\binom {20}2+ \binom{20}4-…-\binom{20}{18}+\binom{20}{20}$

$\binom {20}0 -\binom {20}2+ \binom{20}4-...-\binom{20}{18}+\binom{20}{20}$ The question specifically gives intervals in which the answer is, but it's probably assumed that you should calculate the ...
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2answers
35 views

Multiplication Principle and Inclusion-Exclusion: $2^n = \sum_{i = 0}^n (-1)^i \binom{n}{i} \binom{2n - 2i}{n - 2i}$

I began to compose an unnecessarily complicated answer to this question: If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want ...
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0answers
18 views

Finding the term number and coefficient of a term containing $a^0$

Find the term number and the coefficient of the term containing $a^0$. $(a + a^{-2})^{12}$ My thought process so far is: That the binomial expansion to the $12$th power will have $13$ terms, ...
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1answer
47 views

prove $\sum \limits_{k=1}^n A(n,k){x+k-1 \choose n}=x^n$

A descent in the permutation $\sigma = a_1 \cdots a_n \in S_n$ is an index $i\in[n-1$] for which $a_i > a_{i+1}$. Let A(n, k) be the number of permutations of $[n]$ with $k-1$ descents where $n ...
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2answers
33 views

Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
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1answer
28 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
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5answers
123 views

How to prove $\sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0$

I would like to prove that: \begin{equation*} \sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0;~k\geq0 ; n\geq1. \end{equation*} Can any one help me how to do that? Thanks
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2answers
34 views

Find a formula for the binomial coefficients of the Macluarin series for $\frac{1}{(1+x)^{1/2}}$

The Maclaurin series for $\frac{1}{(1+x)^{1/2}}$ is \begin{equation*} 1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}...~. \end{equation*} I can't figure much out other than it ...
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1answer
25 views

The biggest binomial coefficient in $(n+\frac 1n)^n$ if the product of the fourth member in the expansion and the fourth from back member is 14400

I'm stuck on this one. I am not expected to know how to solve cubic equations so this gets even more confusing, as i get $\binom n 3 = 120$. So I can't even calculate $n$. Is there a way to go around ...
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0answers
15 views

Beta functions and Pascal Pyramids

I've looked over this paper and there was something about the limits presented that looked rather familiar. Then I thought, hey, those look like beta functions: \begin{align} ...
2
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1answer
23 views

Are binomial series multiplicative in their bases?

In $ℚ[Z]$, by $Z \choose k$ denote the polynomial $${Z \choose k} = \frac{1}{k!}·\prod_{i=0}^{k-1} (Z-i),$$ so that ${Z \choose k}(n) = {n \choose k} = \frac{n!}{k!·(n-k)!}$ in $ℚ$. Now, in the ring ...
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2answers
31 views

Enough Information? (Linear Algebra over Finite Fields)

This problem works over the field $\mathbb{F}_p$. Suppose $p$ is a prime and the $i\in I$ index the set of $p$ vectors $v_i$. Fix $k$ an integer with $1\leq k<p$. Let $v_i$ have the following ...
2
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1answer
43 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
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0answers
16 views

Finding the Value of Ties

THE CHALLENGE If "Winning all the time = 1" "Losing all the time = -1" "Tie all the time = 0" What value would "Winning HALF the time" be? Is there an error in my original variables that would help ...
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1answer
29 views

q-binomial Identity

Unfortunately I am not able to solve the following problem: I tried finding a bijection similar to the prove of this binomial identity: $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$ ...
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1answer
27 views

Probability and Statictic / Binomial

The cost of a trial conducted in the research and development center of an industrial establishment is known to be 1 million dolars. If the test is negative, in addition to this a new trial is ...
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1answer
41 views

Generating the Nth combination of a binomial coefficient

I'm designing a protocol, and need a bit of help. I am able to neatly condense the problem I am having into a allegory, I hope it doesn't sound too contrived. Alice has flipped a coin ...
2
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1answer
31 views

Questions on integer-valued polynomials

An integer-valued polynomial or numerical polynomial is a polynomial $f \in \mathbb Q[x]$ with the property that $f(\mathbb Z)\subseteq \mathbb Z$. The set of numerical polynomials forms a subring ...
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30 views

Proving a sequence of numbers in binomial

Consider the set $P_r={n\choose r}p^r(1-p)^{n-r}$ Prove that: $$\sum_{r=1}^nrP_r=np$$ By far I attempted: $$\sum_{r=1}^nr{n\choose r}p^r(1-p)^{n-r}=\sum_{r=1}^nn{n-1\choose r-1}p^r(1-p)^{n-r}$$ ...
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3answers
54 views

Proving that $r{n \choose r}=n{n-1\choose r-1}$

For proving that: $r{n \choose r}=n{n-1\choose r-1}$ I attempted it with: $r{n\choose r}=\frac{rn!}{r!(n-r)!}=\frac{n!}{(r-1)!(n-r)!}$ $n{n-1\choose ...
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5answers
104 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
3
votes
2answers
88 views

Formal power series coefficient problem

Find the coefficient of: $[x^{33}](x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}$ I have figured out that I need to use this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ But I ...
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2answers
59 views

In how many ways can you split six persons in two groups?

In how many ways can you split six persons in two groups? I think that I should use the binomial coefficient to calculate this but I dont know how. If the two groups has to have equal size, then ...
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19 views

Closed form for $\sum _{k=r}^s \binom{n}{k}$

The cardinality of the Powerset is $2^n$. Looking for $\sum _{k=r}^s \binom{n}{k}$, Mathematica gives $$\sum _{k=r}^s \binom{n}{k}=\binom{n}{r} \, _2F_1(1,r-n;r+1;-1)-\binom{n}{s+1} \, ...
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2answers
78 views

Roots of a polynomial whose coefficients are ratios of binomial coefficients

Prove that $\left\{\cot^2\left(\dfrac{k\pi}{2n+1}\right)\right\}_{k=1}^{n}$ are the roots of the equation $$x^n-\dfrac{\dbinom{2n+1}{3}}{\dbinom{2n+1}{1}}x^{n-1} + ...
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1answer
107 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...
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1answer
25 views

Show that $(m-1)^{n-1}*n^{m-2}=\binom{n+m-1}{n}$ [closed]

I need help proving that $(m-1)^{n-1}*n^{m-2}=\binom{n+m-1}{n}$.
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1answer
44 views

Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that ...
6
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4answers
93 views

$\binom{n}{r}$ versus $^n\mathrm{C}_r$ : which notation is more used?

I know that the notation $\binom{n}{r}$ is more standard to use since we have a $\LaTeX$ command for it while there is no such thing for $^n\mathrm{C}_r$. Now, I'm wondering which notation do people ...
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1answer
44 views

binomial distributions and their transforming (6.37-6.39)

I'm lost and frustrated. I don't know how the author (Karl Sigmund; The Calculus of Selfishness) transforms 6.37 in the book pages imaged below: $$ P_y = \sigma w^{N-1} + ...
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1answer
18 views

Approximation Error of Stirlings Formula

Stirlings Approximation : $n! \approx \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$. So $100!$ has an approximate percentage error of about $\frac{100}{12n} = \frac{1}{12}$. Using this information, how does ...
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1answer
42 views

prove an identity involving beta function and gamma function

We know that $B(p,q)=\Gamma(p)\Gamma(q)/\Gamma(p+q)$ where $p, q>0$, and $B(p,q)$ is related to binomial coefficients if one of $p,q$ is an integer. I want to prove the following identity. ...
2
votes
2answers
59 views

Trinomial Theorem for negative exponents

I just learned of binomial theorem for negative integers (or in that case any real $n$). Does such a theorem exist for the trinomial theorem $$(a+b+c)^n$$ and has there been work done? I would think ...
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2answers
82 views

The value of ${\sum_{k=0}^{20}}(-1)^k\binom{30}{k}\binom{30}{k+10}$

$\newcommand{\b}[1]{\left(#1\right)} \newcommand{\c}[1]{{}^{30}{\mathbb C}_{#1}} \newcommand{\r}[1]{\frac1{x^{#1}}}$ The value of $$\sum_{k=0}^{20}(-1)^k\binom{30}{k}\binom{30}{k+10}$$ It is also the ...
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2answers
45 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
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2answers
53 views

Proving a rule of combination [duplicate]

I have this question: Considering $$(1+x)^n(1+x)^n=(1+x)^{2n}$$ and show that: $${n \choose 0}^2+{n \choose 1}^2+{n\choose 2}^2+....+{n\choose n}^2={2n\choose n}$$ I have attempted so far: ...
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2answers
38 views

show $\sum_{k=0}^n {k \choose i} = {n+1 \choose i+1}$

show for n $\geq i \geq 1 : \sum_{k=0}^n {k \choose i} = $ ${n+1} \choose {i+1}$ i show this with induction: for n=i=1: ${1+1} \choose {1+1}$ = $2 \choose 2$ = 1 = $0 \choose 1$ + $1 \choose 1$ = ...
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2answers
64 views

Find the coefficient of $x^6$ [closed]

I am practicing the Binomial theorem but now I am stuck over this question. Find the coefficient of $x^6$ in $(1 + x^2 + x^4 + x^6)^{20}$ Please help. Thank you
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4answers
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Why the $GCD$ of any two consecutive fibonnaci numbers is $1$?

Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof. I did the following: $$F_n=F_{n-1}+F_{n-2} \\ ...
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0answers
24 views

how to estimate $\sum (n,i)$?

How to estimate $\sum\limits_{i=-A}^{A} \binom{n}{i}$ ? This probably relates to the central limit theorem and the proof of it. But I want good estimates, also for small values (not just limits to ...
2
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0answers
22 views

Equation with $q$-binomial coefficients [migrated]

Let $d\ge2$, and let $q$ be a power of a prime. As usual, define $N(d,q)=\sum_{k=0}^d{d\choose k}_q$. I wonder if there are $d$ and $q$ as above such that $1+N(d,q)=q^{d+1}$. (If the answer is ...
3
votes
2answers
93 views

Sum of series with binomial

How to calculate $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{2n}{2^{2n}(2n-1)}$$ ? I tried to use residues, generating function, combinatorics formulas, but unsuccessfully.
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1answer
45 views

Generalisation of Binomial Coefficient (Combinatorics on words)

So, when trying to find subwords from a bigger word: $\binom{abracadabra}{ab} = 5$ with $ABracadabra$, $AbracadaBra$, $abrAcadaBra$, $abracAdaBra$, $abracadABra$. I have noticed that it doesn't go ...
0
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2answers
47 views

$\big(1+\frac {C_0}{C_1}\big) \big(1+\frac {C_1}{C_2} \big) \cdots \big(1+\frac {C_{n-1}}{C_n} \big) =?$

If $(1+x)^n =~~ ^nC_0 ~+~ ^nC_1~x ~+~ ^nC_2 x^2~+~\cdots+~ ^nC_n x^n $, then the value of : $$\bigg(1+\dfrac {C_0}{C_1}\bigg) \bigg(1+\dfrac {C_1}{C_2} \bigg) \cdots \bigg(1+\dfrac ...
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0answers
37 views

How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
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3answers
25 views

Calculating the sum of coefficients for binomial expression

The question is: Calculate the sum of the coefficients of $(a-b)^{250}$. My reasoning was that we can take the example of $(a-b)^2$, which would have the coefficients of $1$, $-2$, and $1$, ...
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0answers
45 views

Mean and variance of a random variable whose distribution is Pr {X = k} = Binomial [N-k, n] *Binomial [k-1,n] /Binomial [N, 2n+1] for k=n+1, …, N-n

What is the mean and variance of the random variable whose distribution function is $P(X = k)=$ $\binom{N-k}{ n} \times \binom{k-1}{n} \over \binom{N}{ 2n+1}$ for $k=n+1, ..., N-n$? Progress: ...