Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

learn more… | top users | synonyms (1)

4
votes
1answer
108 views

Squarefree binomial coefficients.

At $n=23$, all binomial coefficients are squarefree. Is this the largest value for $n$ where this is the case? Edit A plot up to $n=50$: A plot up to $n=500$: plotted against $n+1$ and ...
4
votes
4answers
144 views

Evaluating Sums Algebraically or Combinatorially

Consider (1) $$\sum_{k=0}^{n}\binom{n}{k}2^{k-n}$$ (2) $$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$ These sums appear too difficult (in my mind) to evaluate combinatorially. What are some ...
4
votes
3answers
363 views

Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Consider the following identity: \begin{equation} \sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1} \end{equation} Consider the set $S$ of size $2n-2$. We partition $S$ into two sets $A$ ...
4
votes
2answers
733 views

Trying to prove that $p$ prime divides $\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$

So I'm trying to prove that for any natural number $1\leq k<p$, that $p$ prime divides: $$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$$ Writing these choice functions in ...
4
votes
1answer
229 views

Inequality involving sums of fractions of products of binomial coefficients

Let $n\in\mathbb{N}$. For $0\le l\le n$ consider \begin{equation} b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.} \end{equation} Do you know a technique how ...
4
votes
1answer
200 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
4
votes
2answers
464 views

Limit of alternating sum with binomial coefficient

I need to find a limit, or approximation for $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(a+bk)$ for, say, an $a,b\in (0,10)$. It is not so important what values $a$ and $b$ have. It would be ...
3
votes
1answer
70 views

Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$ [duplicate]

I'm trying to prove this combinatorially. $$k\binom{n}{k} = n\binom{n-1}{k-1}$$ I know the first step is to relate a question to the equation. My question was if you have $n$ friends how many ways can ...
3
votes
1answer
100 views

Proof sought for a sum involving binomials that simplifies to 1/2

A proof of: $$\begin{align*}(1/2)^{2m+1} \sum_{k=0}^{m} \binom{m}{k} \sum_{j=0}^{k} \binom{m+1}{j} = \frac{1}{2} \end{align*} $$ Conjecture based on the following Maple code: ...
3
votes
2answers
75 views

$2^n=C_0+C_1+\dots+C_n$

Could anyone give me hints for this one? $2^n=C_0+C_1+\dots+C_n$ Thats all I can say and I know tricks like integrating or differentiating both side and then put $x=1$ or what ever we need. but ...
3
votes
1answer
98 views

Simplest proof that $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$

The inequality $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$ is very useful in the analysis of algorithms. There are a number of proofs online but is there a particularly elegant and/or simple proof ...
3
votes
2answers
83 views

For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$

Let $A:=\sum_{k=0}^{\infty}x^{2^k}$. For what $n$ is it true that $(A+1)^n+A^n\equiv1\mod2$ (here we are basically working in $\mathbb{F}_2$.) The answer is all powers of 2, and it's fairly simple ...
3
votes
2answers
228 views

proving a sum of binomial coefficients

How can i prove that $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}=2^{2n-1}$ I tried using induction and pascal's identity but it didn't help me.
2
votes
1answer
68 views

Why is this formula for $(2m-1)!!$ correct?

Numerically calculating the sum of the squares of the $m$th row of Pascal's triangle, I found that for at least the first $10$ or so cases $$\sum_{i=0}^m \binom{m}{i}^2=\frac{(4m-2)!!!!}{m!}$$ Where ...
2
votes
2answers
1k views

Sum of square binomial coefficients [duplicate]

Please feel free to close this is necessary as I didn't see exactly this question (some variations that I tried but didn't seem to apply. Prove: $$\sum_{k=0}^{n}{\binom{n}{k}^2}=\binom{2n}{n}$$ I ...
2
votes
2answers
105 views

Prove the following relation:

I must prove the relation $$\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k}=2\sum_{k=0}^n\binom{n+k}{k}\frac1{2^k}.$$ I got this far before I got stuck: $\begin{eqnarray*} ...
2
votes
2answers
222 views

Combinatorial Proof for a $ p\mid\binom{p}{k} \ \ \ \ \ 0<k<p$ .

I'm looking for a combinatorial proof to the following statement: $$ p\mid\binom{p}{k} \ \ \ , \ \ 0<k<p \ \ \ \ \ \ \text{and} \ \ p \ \text{is prime}.$$ Thank you.
1
vote
1answer
147 views

Inequality for binomial coefficients

Let $m \leq n, n \leq N$ and $0\leq k \leq m$. I am wondering what is the dependence of $n$ and $N$ that for all $m, k$ $$ \frac{{N-m \choose n-k}}{{N \choose n}}\leq 1. $$ Thank you for your help.
0
votes
3answers
1k views

Binomial coefficients (1/2, k)

I don't understand questions that involve a binomial expression where you have a fraction choose k or a negative number choose k. I understand and am able to do it when there are no fractions and they ...
10
votes
3answers
299 views

Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$

I try to compute the following limit: $$\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$$ I'm interested in finding some reasonable ways of solving the limit. I don't find any ...
10
votes
2answers
331 views

The integral $\int_0^1 \frac{(x+1)^n-1}{x} dx$

I know that the integral $\int_0^1 \frac{(x+1)^n-1}{x} dx,$ for $n \in \mathbb{Z}^+$, can be evaluated by expanding the numerator with the binomial theorem and integrating term by term. You get the ...
9
votes
3answers
161 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..?
8
votes
2answers
656 views

Binomial coefficients: how to prove an inequality on the $p$-adic valuation?

In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used: $$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...
6
votes
1answer
192 views

$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof

Is there a simple combinatorial proof for the following identity? $$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$ where $m,j \geq 0$, $k \geq n \geq 0$.
5
votes
2answers
464 views

Odd Binomial Coefficients?

By Newton's Formula: $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k $$ Proof that every $\dbinom{n}{k}$ is odd if and only if $n=2^r-1$. I have already shown that if $n$ is of the form $2^r-1$, ...
5
votes
0answers
184 views

Partial sum over $M$, of ${m+j \choose M} {1-M \choose m+i-M}$?

Is there (likely to be) any formula for $$ \sum_{m' \geq m} {m+j \choose m'} {B - m' \choose m+i-m'} $$ ? I am mainly interested in the case $B=1$ (or a prime power), if that's any ...
5
votes
3answers
3k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
4
votes
5answers
201 views

Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$

I've seen the identity $\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ used here recently. I checked for proofs here ...
4
votes
1answer
266 views

Divisibility of binomial coefficient by prime power - Kummer's theorem

Let's say we have binomial coefficient $\binom{n}{m}$. And we need to find the greatest power of prime $p$ that divides it. Usually Kummer's theorem is stated in terms of the number of carries you ...
4
votes
1answer
231 views

Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$

Prove that $$\binom{2n}{n}\ge\dfrac{2^{2n-1}}{\sqrt{n}}$$ By the way: I have see $$\binom{2n}{n}\ge\dfrac{4^n}{2n}=\dfrac{2^{2n-1}}{n}$$ proof: Applying the binomial theorem ...
4
votes
4answers
230 views

Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
4
votes
1answer
53 views

$\frac{1}{4^n}\binom{1/2}{n} \stackrel{?}{=} \frac{1}{1+2n}\binom{n+1/2}{2n}$ - An identity for fractional binomial coefficients

In trying to write an answer to this question: calculate the roots of $z = 1 + z^{1/2}$ using Lagrange expansion I have come across the identity $$ \frac{1}{4^n}\binom{1/2}{n} = ...
4
votes
3answers
947 views

Distributing identical objects to identical boxes

We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
4
votes
2answers
571 views

Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$

This is a question from my first proofs homework and I am confused about the combinatorial argument aspect. I already did the algebraic proof. I think I am supposed to put into words what both sides ...
4
votes
3answers
461 views

Binomial Coefficients in the Binomial Theorem - Why Does It Work Question

to keep it simple: Given $(a+b)^3=\binom{3}{0}a^3+\binom{3}{1}a^2b+\binom{3}{2}ab^2+\binom{3}{3}b^3$ Could you please give me an intuitive combinatoric reason to why the binomial coefficients are ...
4
votes
2answers
624 views

Why does $\sum\limits_{i=0}^k {k\choose i}=2^k$ [duplicate]

Possible Duplicate: Proving a special case of the binomial theorem Can anyone explain to me why $$\sum\limits_{i=0}^k {k\choose i}=2^k\,?$$ Thanks in advance
4
votes
2answers
506 views

How to find $\sum_{k=1}^n 2^kC(n,k)$?

How to find the sum of series, $$\sum_{k=1}^n 2^kC(n,k)$$
4
votes
3answers
1k views

Estimating a certain row of Pascal's triangle

I need to calculate all the numbers in a certain row of Pascal's triangle. Obviously, this is easy to do with combinatorics. However, what do you do when you need to estimate all the numbers in, say, ...
3
votes
2answers
51 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
votes
4answers
127 views

Find $\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$

Find $$\sum_{r=0}^{2n}\frac{r}{r+2}\binom{2n}r\;.$$ I got $$\frac{2^{2n+1}(2n^2+n+1)-1}{(2n+1)(2n+2)}$$ but the answer is $$\frac{2^{2n+1}(2n^2-n+1)-2}{(2n+1)(2n+2)}$$ Thanks for the help...
3
votes
1answer
107 views

Find the constant $c$ in the equation $\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$

Find the constant $c$ in the equation $$\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$$ I've tried to use this asymptotics $$C_n^k \sim \frac{n^m}{m!} \sim e^{m\ln n ...
3
votes
3answers
130 views

Stirling Binomial Polynomial

Let $\{\cdot\}$ denote Stirling Numbers of the second kind. Let $(\cdot)$ denote the usual binomial coefficients. It is known that $$\sum_{j=k}^n {n\choose j} \left\{\begin{matrix} j \\ k ...
3
votes
1answer
124 views

Combinatorial Proof of a Binomial Identity

$$\sum_k {m\choose k} {n \choose k} = {m+n \choose n}$$ In this identity we seem to be choosing subsets that do $\it not$ contain k of type m and type n for all possible k. In the style of ...
3
votes
1answer
131 views

Is $\sum_{k=1}^{n} k^{k-1} (n-k)^{2n-k} \binom{n}{k} \sim\frac{n^{2n}}{2\pi} $?

How can you compute the asymptotics of $$T=\sum_{k=1}^{n} k^{k-1} (n-k)^{2n-k} \binom{n}{k}\;?$$ This is related to Asymptotics of sum of binomials . I attempted to simply use Stirling's ...
3
votes
1answer
150 views

How do i reduce this expression of binomial coefficients

I was solving a problem and am stuck with this expression. Any leads on how can I simplify this expression? $$\frac{{\sum\limits_{x=Q}^{N-P+Q} (x-Q) \binom{x}{Q} ...
3
votes
4answers
590 views

Trying to find $\sum\limits_{k=0}^n k \binom{n}{k}$ [duplicate]

Possible Duplicate: How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$? $$\begin{align} &\sum_{k=0}^n k \binom{n}{k} =\\ &\sum_{k=0}^n k ...
3
votes
2answers
134 views

Why is the binomial coefficient related to the binomial theorem?

The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of ...
3
votes
2answers
318 views

Number of nonnegative integral solutions of $x_1 + x_2 + \cdots + x_k = n$

To find all solutions greater than or equal to $1$ of a linear equation in the form $$x_1+x_2+x_3+\cdots+x_k=n ,$$ the number of them is $\binom{n-1}{k-1}$. If I need all solutions to be greater or ...
3
votes
3answers
844 views

How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?

I want to prove that the binomial coefficient ${n \choose k}$ for $n \ge k$ is a monotonically nondecreasing sequence for a fixed $k$. How do I do this?
3
votes
1answer
228 views

binomial expansion

$\displaystyle \binom{n}{k}=\binom{n-1}{k} + \binom{n-1}{k-1}$ $\displaystyle \left(1+x\right)^{n} = \left(1+x\right)\left(1+x\right)^{n-1}$ how do I use binomial expansion on the second equations ...