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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Identity for convolution of central binomial coefficients

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
14
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3answers
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Proof that a Combination is an integer

From its definition a combination $(^n_k)$, is the number of distinct subsets of size k from a set of n elements. This is clearly an integer, however I was curious as to why the equation ...
8
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6answers
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Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$

I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= ...
11
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4answers
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Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
4
votes
2answers
82 views

Evaluating Combination Sums

Evaluate $$\sum_{k=0}^n{n+k\choose 2k} 2^{n-k}$$ So im not really sure how to begin with this. I would imagine we start with dividing out $2^{n}$, but not really sure much past that
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3answers
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Proof of a combinatorial identity: $\sum_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$ [duplicate]

Possible Duplicate: Identity involving binomial coefficients This was part of a homework assignment that I had, and I couldn't figure it out. Now it is bugging me. Can anyone help me? ...
7
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10answers
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How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?

I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am ...
20
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2answers
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How to reverse the $n$ choose $k$ formula?

If I want to find how many possible ways there are to choose k out of n elements I know you can use the simple formula below: $$ \binom{n}{k} = \frac{n! }{ k!(n-k)! } .$$ What if I want to go the ...
2
votes
3answers
480 views

Vandermonde's Identity : Summations with binomial coefficients

(Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Pr, 8th ed by Sheldon Ross. Can someone help me solve this equation? How to prove that ...
10
votes
3answers
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Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates?
25
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2answers
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How do I count the subsets of a set whose number of elements is divisible by 3? 4?

Let $S$ be a set of size $n$. There is an easy way to count the number of subsets with an even number of elements. Algebraically, it comes from the fact that $\displaystyle \sum_{k=0}^{n} {n ...
20
votes
6answers
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prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer

Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared. My thought process: The numerator is the product of the first n even ...
2
votes
3answers
620 views

Counting subsets containing three consecutive elements (previously Summation over large values of nCr)

Problem: In how many ways can you select at least $3$ items consecutively out of a set of $n ( 3\leqslant n \leqslant10^{15}$) items. Since the answer could be very large, output it modulo $10^{9}+7$. ...
23
votes
7answers
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Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
15
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8answers
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Proving $\sum\limits_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$

Show that $$\sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$$ So for odd $n$ we have an even number of terms. So $\binom{n}{k} = \binom{n}{n-k}$ which have opposite signs. Thus the sum is $0$. For even $n$ ...
5
votes
4answers
438 views

Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$ [duplicate]

I want to know if I can get some help with this proof. I tried, but I just cannot seem to get $2^{k}$. It states that, For $k \in \mathbb{Z}_{\ge 0}$, $$\sum^{k}_{m=0}\binom{k}{m} = 2^k$$ Thank ...
18
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6answers
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Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$

I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: ...
11
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1answer
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Combinatorial proof of summation of $\sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Can you guys help me prove this? There is a way of proving this logically but I was hoping to find a more "mathematical" proof, if possible. $$\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n ...
7
votes
5answers
684 views

Proof of $\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1}$?

How do I prove that $$\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1} \>?$$ I saw this in a book discussing generating functions.
7
votes
4answers
558 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
3
votes
5answers
284 views

Proving the total number of subsets of S is equal to $2^n$

Student here! Just reading Liebecks Introduction to pure mathematics for fun and I made an attempt at proving the total number of subsets of S is equal to $2^n$. I realized that the total number of ...
1
vote
2answers
358 views

simple binomial theorem proof

I am trying to prove this binomial statement: For $a \in \mathbb{C}$ and $k \in \mathbb{N_0}$, $\sum_{j=0}^{k} {a+j \choose j} = {a+k+1 \choose k}.$ I am stuck where and how to start. My steps ...
23
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6answers
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Beautiful identity: $\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$

Let $m,n\ge 1$ be two integers. Prove that $$\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$$ where $\delta_{mn}$ stands for the Kronecker's delta. Note: I put the tag "linear ...
5
votes
4answers
232 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
5
votes
4answers
223 views

Proving $\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$

Prove that $$\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$$ by computing the coefficient of $z^M$ in the identity $$(1 + z + z^2 + \cdots ) \cdot \frac{1}{(1-z)^{k+1}} = \frac1{(1-z)^{k+2}}.$$ I ...
9
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4answers
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$C(n,p)$: even or odd?

Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value? ($p\lt n$, $p$ and $n$ are positive integers)
4
votes
1answer
581 views

Why is $n\choose k$ periodic modulo $p$ with period $p^e$?

Given some integer $k$, define the sequence $a_n={n\choose k}$. Claim: $a_n$ is periodic modulo a prime $p$ with the period being the least power $p^e$ of $p$ such that $k<p^e$. In other words, ...
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2answers
666 views

Induction proof concerning a sum of binomial coefficients: $\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$

I'm looking for a proof of this identity but where j=m not j=0 http://www.proofwiki.org/wiki/Sum_of_Binomial_Coefficients_over_Upper_Index $$\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$$
3
votes
2answers
310 views

lacunary sum of binomial coefficients

I am sure there must be a known estimate for sums of the form: $$ S_d(n)=\sum_{j=0}^n \binom{dn}{dj} $$ where $d\ge 1$. For $d=1$, the answer is obvious. For $d=2$, the answer is here: Sum with ...
40
votes
5answers
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$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...
16
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7answers
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Prime dividing the binomial coefficients

It is quite easy to show that for every prime $p$ and $0<i<p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the ...
13
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5answers
527 views

Combinatorially showing $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$

I am trying to show that $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$. I found that using stirling's approximation, I can get: $$ \lim_{n\to \infty}{\frac{2n\choose n}{4^n}}= \lim_{n\to ...
3
votes
2answers
176 views

Proving Binomial Idenity without calculus

How to establish the following identities without the help of calculus: For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$ and $$\sum_{0\le r\le ...
7
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1answer
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Bounds on $\sum_{k=0}^{m} \binom{n}{k}x^k$ and $\sum_{k=0}^{m} \binom{n}{k}x^k(1-x)^{n-k}, m<n$

I've read this interesting article by Woersch (1994) dealing with approximation of binomial coefficients (rows of Pascal's triangle). I'm just wondering if similar bounds exist for partial binomial ...
6
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1answer
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Partial sum of rows of Pascal's triangle

I'm interested in finding $$\sum_{k=0}^m \binom{n}{k}, \quad m<n$$ which form rows of Pascal's triangle. Surely $\sum\limits_{k=0}^n \binom{k}{m}$ using addition formula, but the one above ...
5
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4answers
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Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$

I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of ...
5
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1answer
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Asymptotics for a partial sum of binomial coefficients

Good afternoon, I would like to ask, if anyone knows how to evaluate a sum $$\sum_{k=0}^{\lambda n}{n \choose k}$$ for fixed $\lambda < 1/2$ with absolute error $O(n^{-1})$, or better. In ...
7
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3answers
686 views

Combinatorial proof for two identities [duplicate]

Does exist a combinatorial proof for the following two identities ? $\sum_{k = 0}^{n} \binom{x+k}{k} = \binom{x+n+1}{n}$ $\sum_{k = 0}^{n} k\binom{n}{k} = n2^{n-1}$ I know how to derive the ...
6
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2answers
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Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$

I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$ I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
5
votes
4answers
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How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
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2answers
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How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
0
votes
5answers
149 views

Prove by induction: $2^n = C(n,0) + C(n,1) + \cdots + C(n,n)$ [duplicate]

This is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. $$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$ Prove this statement is true for all $n ...
17
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3answers
392 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
17
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6answers
499 views

How to prove $\sum\limits_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$?

How do I prove the following identity directly? $$\sum_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$$ I thought about using the binomial theorem for $(x+a)^n$, but got stuck, because I realized ...
15
votes
2answers
947 views

Combinatorial interpretation of Binomial Inversion

It is known that if $f_n = \sum\limits_{i=0}^{n} g_i \binom{n}{i}$ for all $0 \le n \le m$, then $g_n = \sum_{i=0}^{n} (-1)^{i+n} f_i \binom{n}{i}$ for $0 \le n \le m$. This sort of inversion is ...
14
votes
5answers
882 views

Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially

$$\text{Evaluate } \sum_{k=1}^n k^2 \text{ and } \sum_{k=1}^{n}k(k+1) \text{ combinatorially.}$$ For the first one, I was able to express $k^2$ in terms of the binomial coefficients by ...
14
votes
4answers
503 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
12
votes
6answers
841 views

Proving a binomial sum identity

Mathematica tells me that $$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$ Although I have not been able to come up with a proof. Proofs, hints, or references are all ...
11
votes
3answers
308 views

Show that $\sum_{k=0}^n\binom{2n}{2k}^{\!2}-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$

How can I prove this $$ \sum_{k=0}^n\binom{2n}{2k}^2-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^2=(-1)^n\binom{2n}{n}? $$ Maybe can we expand $$ f(x)=(1+x)^{2n}? $$ Thank you.
7
votes
2answers
705 views

Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$

We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$