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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How to expand $(x+y)^{-n}$?

How to expand $(x+y)^{-n}$ by binomial theorem, where $n$ is a positive integer? Is there any limitation for $x$ and $y$? If it can be expanded, how to compute the coefficients? Many thanks in ...
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1answer
26 views

Approximating a binomial sum over a simplex

For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with. Can we do something ...
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1answer
26 views

Compute convolution

Let $a>0$ and $a_1>0$ be real numbers. Using the convolution theorem I have shown a following identity: \begin{equation} \sum\limits_{l=1}^n \binom{a_1}{n-l} \frac{1}{l} \binom{a}{l} = -\gamma ...
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1answer
30 views

Express $\binom{n+2}{k}$ according to $\binom{n}{k}$

I've just begun studying binomial coefficient and I'm trying to express $\dbinom{n+2}{k}$ according to $ \dbinom{n}{k}$. With this result I have to conclude that $\dbinom{2n}{2k}, \dbinom{2n+1}{2k}$ ...
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1answer
32 views

To prove for every integer $k >1 $ , $2+2^k \choose r$ is even for all $2 < r \le n=1+2^{k-1}$ , without Lucas' theorem

Can we prove by induction that for every integer $k >1 $ , $2+2^k \choose r$ is even for all $2 < r \le n=1+2^{k-1}$ ? Or by some divisibility properties of Binomial co-efficients ? I wanted to ...
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1answer
56 views

Use of Multinomial theorem.

I have the next identity which I want to prove. $$(\sum_{j}k_j^2)^{s} = \sum_{b_1+\ldots+ b_n =s} \prod_j k_j^{2b_j}$$ Obviously I need to use the Multinomial theorem, but how to procceed from ...
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3answers
58 views

How can I compute $\sum\limits_{k = 1}^n \binom{n - 1}{k - 1}$?

I know what $n \choose k$ equals, but I don't see how that would help me solve the sum of $n - 1 \choose k - 1$ from $k = 1$ to $n$. Is there any special trick I should know?
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1answer
342 views

How to deal with this double summation?

I'm stuck with the proof of this result: $$2^n = \sum_{t=-\frac{n-1}{2}}^{\frac{n-1}{2}} \binom{n+1}{\frac{n+1}{2} + t} \sum_{k=\vert t \vert}^{\frac{n-1}{2}} \binom{\frac{n-1}{2}+k}{k} ...
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1answer
55 views

To prove ${2p - 1 \choose p } \equiv 1 \pmod{p^2}$ without using Wolstenholme's theorem

How to prove that ${2p - 1 \choose p} \equiv 1 \pmod{p^2}$ ? I don't want to use Wolstenholme's theorem; but one might use $p|{p \choose k} , 1 \le k \le p - 1$ , and $(p - 1)! \sum_{k = 1}^{p - 1} ...
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1answer
75 views

What are all the even positive integers $n$ such that for $1 \le r \le n-1 $ , $2n \choose r$ is odd for exactly one $r$ ?

Motivated from this To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$ ; what are all the even positive integers $n$ such that for $1 \le r ...
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1answer
15 views

Production process testing - binomial probabilities

A process produces faulty items with probability 0.04. Samples of 100 items are taken from batches at random and if there are less than 5 faulty items in the sample then the batch is accepted; ...
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3answers
178 views

Show that there are infinitely many integers such that $ \binom{m}{n-1} = \binom{m-1}{n} $

This question comes from the 1st Brazilian's IMO TST of 2004. I have found no solutions of it online, though I have developed one. After getting to $ mn = (m-n)(m-n+1) $, my solution relies on the ...
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0answers
35 views

Simpler expression for binomial sum

Is there any closed expression for the following sum: $$\sum_{i=0}^{l-k} \binom{n-l}{i} \binom{l-k}{i} \binom{l-i}{k}$$ where $ k<l < n/2$?
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2answers
140 views

What does 9P2 mean, and how would one solve it?

We are studying "Sequences, Series, and Probability" and it likely related to binomial theorem and pascals triangle. I've a test tomorrow morning, and if I can't figure this out soon, I'm likely to ...
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1answer
113 views

Mean value of minimum of binomial variables

Let $X,Y$ be two independent random variables with binomial distribution: $B[4n,p]$ with $p=0.5$. Let $M=\min(X,Y)$. What is the expectation of $M$? This question seems related but it has no answer: ...
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2answers
41 views

Some binomial equality

I am trying to prove the following equality $$ \sum_{r=k}^{n}\binom{2n+1}{2r+1}\binom{r}{k}=\binom{2n-k}{k}2^{2n-2k}~~;~k\le n. $$ I noticed that for $k=0$ it becomes $$ ...
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1answer
44 views

$k$ divides $\binom{kn}{n}$

For positive integers $k,n$ , is it true that $k$ divides $\binom{kn}{n}$? I can write $$\binom{kn}{n}=\frac{(kn)(kn-1)\cdots(kn-n+1)}{n(n-1)\cdots 1}$$ but must the $k$ at the top remain after ...
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2answers
63 views

Legendre polynomial to show identity, can't spot mistake

Using Legendre polynomial generating function \begin{equation} \sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}} \end{equation} Or $$ P_n(x)=\frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] ...
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1answer
123 views

To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$

For which odd integers $n>1$ is it true that $2n \choose r$ where $1 \le r \le n$ is odd only for $r=2$ ? I know that $2n \choose 2$ is odd if $n$ is odd but I want to find those odd $n$ for which ...
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1answer
85 views

Binomial expansion (sort of ) rearrangement

Let $$ x_n=\sum_{i=1}^n\binom{n}{i}y_iz_{n-i} \qquad n=1,\ldots,k $$ For general $k$, can you find an explicit expression for $y_k$ only in terms of $x_1,\ldots,x_k$ and $z_1,\ldots,z_k$? For ...
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1answer
54 views

Can this expression be simplified?

I get a function related to a variable $x$, that is $f(x)=\displaystyle \frac{ \sum_{i=0}^K{ i \binom{n+i}{i} x^i}} {\sum_{i=0}^K{\binom{n+i}{i}x^i}}$, where $K$ and $n$ are positive integers, can ...
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1answer
34 views

Polynomials and difference operator

Let's consider a difference operator $\triangle f(n)= f(n+1)-f(n)$. How to prove that $f$ is a polynomial so that $deg(f) \leq d$ if and only if $\triangle f ^{d+1} =0$. First step of the solution ...
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1answer
179 views

Hard binomial sum [closed]

How to prove this relation? $$\sum_{i=0}^{n}\frac{2^{-2i}\binom{2i}{i}}{n+i+2}=\frac{2^{4n+2}-\binom{2n+1}{n}^2}{(2n+3)2^{2n+1}\binom{2n+1}{n}}$$ That seems difficult!
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4answers
203 views

Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$

A teacher gave this as a homework question, and I have tried but haven't been able to arrive at a solution. $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ Could someone prove it, or at least ...
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1answer
38 views

How to prove elementary identities for binomial coefficients using combinatorial arguments?

I'm in a second year discrete mathematics course, and we have identities like this $$\binom{n}{k}(n-k) = \binom{n-1}{k}n$$ and Pascal's Triangle law. Our professor said that algebraic proofs are ...
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0answers
34 views

How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the ...
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1answer
28 views

How many ways are there to do this experiment?

I don't want the full solution rather a step in the right direction. I believe what I have so far is right but I just would like to verify and know the final basic steps to find out how many ways the ...
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2answers
68 views

Binomial coefficient identity $\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$ [duplicate]

I'm having a bit of problems proving the following: $$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$ I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue. Could anyone ...
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1answer
44 views

length of printout of pascal's triangle

when writing a program, that calculates pascals triangle, I noticed that the printout looks (to me) like an aproximately exponential curve 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 ...
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1answer
43 views

Sum $(1-x)^n$ $\sum_{r=1}^n$ $r$ $n\choose r$ $(\frac{x}{1-x})^r$

The question is to find the value of: $n\choose 1$$x(1-x)^{n-1}$ +2.$n\choose2$$x^2(1-x)^{n-2}$ + 3$n\choose3$$x^3(1-x)^{n-3}$ .......n$n\choose n$$x^n$. I wrote the general term and tried to sum it ...
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2answers
97 views

divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...
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1answer
126 views

proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let $n ≥ 0$ and $k ≥ 0$ be integers. 1) How many bitstrings of length $n + 1$ have exactly $k + 1$ many $1$s? 2) Let $i$ be an integer with $k ≤ i ≤ n$. What is the number of bitstrings of length $n ...
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1answer
41 views

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$ Should I use the formula $C(n,k) = n!/[k!(n-k)!]$? And what is the solution of this problem?
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60 views

Binomial coefficient problem

I still haven't quite realized how to solve binomial coefficient problems like this, can someone show me an elaborated way of solving this? I need to write this expression in a more simplified way: ...
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1answer
34 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
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3answers
75 views

Proof by induction, binomial coefficient

I have to make the following proof: $${\sum\limits_{k=1}^n}{k}{n\choose k} = n2^{n-1}$$ Base case, $n = 1$: $${\sum\limits_{k=1}^{1}}{k}{1\choose k} = 1 = 1\cdot2^0=1$$ Inductive Hypothesis: for ...
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67 views

Proof of non divisibililty of $\binom{n}{r}$ with a prime $p$

I came across this : "It is possible to show that if $p$ is prime, $\binom{n}{r}$ is not divisible by $p$ if and only if the addition $r + (n-r)$, when written in base $p$, has no carries. This means ...
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Prove by induction that the binomial coefficient equals the number of subsets of given size

Prove by induction on $n$ that the binomial coefficient $\begin{pmatrix}n\\m\end{pmatrix}$ is the number of subsets of $I_{n}$ having size equal to $m$. The solution is as follows: So far it can be ...
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5answers
135 views

how to prove: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ [duplicate]

need help to prove this: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ where $n$ is integer $\geq 1$. Question also said taking the derivative of $(1 + x)^n$ would be helpful which I've found ...
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0answers
70 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
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3answers
392 views

Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
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3answers
151 views

The maximum of $\binom{n}{x+1}-\binom{n}{x}$

The following question comes from an American Olympiad problem. The reason why I am posting it here is that, although it seems really easy, it allows for some different and really interesting ...
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5answers
205 views

Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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1answer
42 views

What is $\binom{n}{-k}$?

What is $\binom{n}{-k}$ ? If $n,k\ge0$ In Wikipedia there's a case where $n$ is negative and not $k$ But if Pascal's rule still holds, I get for example for $k=0$; ...
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2answers
41 views

Binomial Expansion where N is negative

Comparing the formula for regular binomial expansion (n>1): $(a+b)^n=a^n + \binom{n}1a^{n-1}b + \binom{n}2a^{n-2}b^2 +...$ to binomial expansion for negative indices, (n<1): $(1+x)^n= 1 + nx + ...
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1answer
62 views

Proof By Induction Using Binomial Coefficients

I'm having a really hard time with this proof by induction: Prove this formula by induction: $1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$. Easy enough, right? Wrong. I have to do it using ...
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1answer
35 views

Finding the coefficient of a power series

How would I find the coefficient of: $[x^{10}]x^6(1-2x)^{-5}$ I know that I can simplify this as follows: $[x^4](1-2x)^{-5}$ and that generally the following formula would be used to solve this: ...
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0answers
46 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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1answer
37 views

Binomial coefficients bounded by entropy exponential

So I'm trying to prove that for $\frac{1}{2}< x \leq 1$ we have $$\sum_{\lceil nx \rceil}^{n}{n \choose k} \leq 2^{nh(x)}$$ I've managed to prove that $$\sum_{0}^{\lfloor nx \rfloor}{ n\choose ...
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4answers
63 views

Limit of quotient of summations involving special binomial coefficients

Find the limit, when $n$ tends to infinity, of $$ \frac{\displaystyle\sum_{k=0}^n\binom{2n}{2k}3^k} {\displaystyle\sum_{k=0}^{n-1}\binom{2n}{2k+1}3^k} $$ Please Help Me to solve the problem ...