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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Number of ways of selecting atleast one book from 9 different books of 10 copies each.

Number of ways of selecting atleast one book from 9 different books of 10 copies each. Let $x_i$ denote the number of copies selected from $i^{\text{th}}$ type of book. $$\sum_{i=1}^9 x_i\le 90$$ I ...
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2answers
51 views

Variation on Vandermonde's identity

How can you show that $$ \binom{2n}{n}^2 = \sum_{m=0}^{n} \binom{2n}{2m} \binom{2m}m \binom{2n-2m}{n-m} $$? I was fooling around with random walks, and apparently both expressions are supposed to be ...
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3answers
78 views

Criticize my math when I attempt to find the coefficient of $x^2y^6$ in the expansion of $(x+2y^2)^5$

So I look around this site and my textbook (Richmond&Richmond, discrete math) and I know I'm in the right direction but I'm also sure I am doing it wrong. Original Question: find the coefficients ...
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67 views

Spread Polynomials Identity (Rational Trigonometry)

Show that if $ n=2l+1 $ is an odd natural number then $$ S_{n}\left( s \right)=s\left( \binom{n}{1}\left( 1-s \right)^{l}-\binom{n}{3}\left( 1-s \right)^{l-1}s+\binom{n}{5}\left( 1-s ...
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1answer
37 views

How to differentiate $\big[\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k\big]$ with respect to $n$ without summing?

I know the answer because of the following derivation: $$ {d\over d n}\left[\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k\right]= {d\over d n}(a+b)^n=(a+b)^n\log{(a+b)} $$ The way of calculating it is to ...
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1answer
26 views

Is there a way to mathematically express the sum of all combinations of a set of constants?

Suppose that I have a set of $n$ constants. I want to find the sums of all product combinations of length $i = n \rightarrow 0$ Using $n=5$ as an example: $C = {a,b,c,d,e} $ $C_5= abcde$ $C_4= ...
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61 views

A combinatorial proof for a bound on diagonal Ramsey numbers

I wish to prove $R(p,p)\leq\frac{2^{2p-2}}{\sqrt{p}}$ combinatorially. I have proved this algebraically through the definition of the binomial coefficient but I would much prefer a proof from ...
3
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5answers
47 views

Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal

Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal. My attempt: $\displaystyle ...
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2answers
41 views

Trouble understanding how this identity is derived: $\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$

$$\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$$ The $-a-1$ is throwing me off. Can anyone help me understand this identity. I have tried letting $m=-a-1$ and then applying the binomial ...
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1answer
85 views

Encyclopedia of Integer Sequences - Formula

I am trying to reproduce the following sequence (https://oeis.org/A062734): ...
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16 views

rows of pascals triangle as powers of 11 in different numeral systems

It's not too difficult to see that (and understand why) in a base n system the ciphers of $(11_n)^k$ are equivalent to the k-th (0-indexed) row of pascals triangle until one of the numbers becomes ...
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1answer
49 views

Formula for a geometric series weighted by binomial coefficients (sum over the upper index):$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$

The binomial sum is $$\sum\limits_{i=0}^n {n\choose i}\ x^i = (1+x)^n,$$ where $\displaystyle{n\choose i}=\frac{n!}{(n-i)!i!}.$ Is there a corresponding formula when you sum over the upper index of ...
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2answers
26 views

Probability that binomial random variable is greater than another

Let $X$ and $Y$ be two independent random variables with respective distributions $B(n+1,\frac{1}{2})$ and $B(n,\frac{1}{2})$. I am trying to determine $\mathbb{P}(X>Y)$. So far, I have written ...
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2answers
55 views

Prove ${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$

I am looking to prove, by induction, the following equality:$${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$$ From Pascal's identity, I know we have that $${n+1 \choose k} = {n \choose k} + {n ...
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1answer
27 views

Complex counting problem

A code has five symbols and each symbol is composed of a letter, a number, and a color. (Ex.{a,3,black} is a symbol) there are $p$ letters, $q$ numbers and $r$ colors to choose from. a ...
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1answer
28 views

Binomial Coefficient Explanaition

Let $n\in\mathbb{N}$ and let $k\in\{0,\ldots,n\}$. Explain why it follows from $$\binom nk=\frac{n}{1}\times\frac{n-1}{2}\times\frac{n-2}{3}\times\cdots\times\frac{n-k+1}{k}$$ that $$\binom ...
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1answer
19 views

Binomial Coefficient Pattern [closed]

Let $n$ $\epsilon$ N and let $k$ $\epsilon$ {0,...,n}. Explain why it follows from ${n \choose k}$ = ${n \choose k-1}$$\frac{n-k+1}{k}$ that ${n \choose k}$ = ...
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27 views

Evaluating a sum on binomial coefficients

I'm reading Casella's and Berger's Statistical Inference. On page 239 they gives a claim that $$\sum_{x=0}^{330}\binom{300+x-1}x\left(\frac{1}{2}\right )^{300}\left (\frac{1}{2}\right )^x\approx ...
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2answers
33 views

Calculation of the limit of the difference of binomial coefficients

This question pertains to harmonic analysis on spheres. Let $H_d$ = {homogeneous, total degree $d$ harmonic polynomials in $\mathbb{C}[x_1,\dots,x_n]$} Given that the Dimension of $H_d = ...
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20 views

Evaluate the sum $\sum_{i=0}^n \binom{n}{i}^2 i^k$

Given a positive integer $k$, is it possible to evaluate the following sum? $$ \sum_{i=0}^n \binom{n}{i}^2 i^k\,\,\,? $$ [I know just for $k=0$ the sum is $\binom{2n}{n} \approx 4^n/\sqrt{n}$..]
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1answer
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Factorials and equivalency

I am not sure if this would be a proper title because I am a bit confused, but I was reading about proving Pascal's Triangle, and there was a proof on here I was following everything that was ...
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3answers
30 views

Binomial coefficients algebra

I am trying to prove $$\binom{n+1}{k} = \binom{n+1}{k-1}\frac{n-k+2}{k}$$ by using the following four equations: \begin{align*} \binom{n + 1}{k} & = \binom{n}{k} + \binom{n}{k - 1}\\ \binom{n + ...
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0answers
31 views

Pascal's triangle induction proof

I am trying to prove $$\binom{n}{k} = \binom{n}{k-1}\frac{n-k+1}{k}$$ for each $k \in \{1,...,n\}$ by induction. My professor gave us a hint for the inductive step to use the following four ...
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2answers
35 views

Coefficient of $x^{n-1}$ in the given expansion

The problem I am facing is that with each term, number of ways to achieve $x^{n-1}$ is increasing, so it is getting very difficult to club all the cases together. Please provide some insight.
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1answer
60 views

Prove the following equality: $ \sum_{i=0} ^n j {n \choose j} = n 2^{n-1} $ [duplicate]

I'd like some help. My first idea was to use induction, but then I get stuck. The base case works just fine, as you'd imagine, and then... If $ \sum_{i=0} ^n j {n \choose j} = n 2^{n-1} \rightarrow ...
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1answer
26 views

On a corollary of Kummer theorem

I faced this problem when I learn Kummer theorem : Let $p$ be a prime number and $m, n$ are two positive integer such that $p\nmid m$ and $p^k\mid n$ for some positive integer $k$. Prove that ...
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On a corrolary of Lucas theorem

In this paper, the author stated that we can use Lucas theorem to prove that $$\binom{ap^f+r}{m}\equiv \binom{r}{m}\ (\text{mod}\ p)$$ where $0\le r, m < p^f$, $a\ge 0$. I don't know how to use ...
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1answer
47 views

Find the remainder when divided by $2017$

Given $2017$ is a prime number. Let $S=\sum_{k=0}^{k=62} \binom{2014}{k}$. Find the remainder when $S$ is divided by $2017$. I am unable to simplify the expression for $S$. Need some hints. ...
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49 views

Why is $\binom{n}{3}p^3\left(1+\frac{3(np)^2}{n}\right) \le \frac{10(np)^5}{n}$

Let $Z = \binom{n}{3}p^3$ and $w=np$. Here $p$ represents a probability. Why is $Z\left(1+ \frac{3w^2}{n}\right) \le \frac{10w^5}{n}$? Edit: There are some other condtions that I didn't think were ...
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188 views

Is there a closed form or approximation to $\sum_{i=0}^n\binom{\binom{n}{i}}{i}$

I tried to calculate the sum $$ \sum_{i=0}^n\binom{\binom{n}{i}}{i} $$ but it seems that all my known methods are poor for this. Not to mention the intimate recursion, that is $$ ...
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1answer
43 views

Divisibility of Binomial Coefficients by a Composite Number [duplicate]

I am aware of proof of divisibility of binomial coefficients of a prime $p$. I've seen it is easy to show that when $0<k<p$ $$\binom{p}{k}\equiv 0 \mod p$$ Can there be anything stronger. ...
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51 views

How do you determine the following coefficient?

$$[x^n]\frac{(1+x)^n}{(1-x)}$$ It seems pretty simple but I can't seem to find it. I tried rewriting it as the product of two sums. $(1+x)^n=\sum_{k=0}^n {n \choose k}x^n$ and ...
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2answers
54 views

What is the sum of binomial-coefficients multiplied by i?

$$\sum_{i=0}^n {n \choose i}i$$ I know this is equivalent to $\sum_{i=0}^n \frac{n!i}{i!(n-i)!}$, but the factorial prevents me from solving this easily.
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1answer
47 views

Algebraic identity involving powers of twin primes

Yesterday, I verified that, if $a$,$b$ and $c$ are real numbers such that $a+b+c=0$, then $$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot\frac{a^2+b^2+c^2}{2}$$ and ...
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2answers
130 views

Subtracting even and odd binomial coefficients?

What number do I get if I subtract the binomial coefficients $n\choose k$ with an even $k$ from those with an odd $k$, where $n$ is fixed? Am I supposed to subtract a binomial coefficient with an even ...
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1answer
32 views

Binomial identity $\sum_{i=0}^{n-1-j}\binom{n}{i+j}\binom{i+j}{j}(-1)^i=\binom{n}{j}(-1)^{n+j+1}$

Let $n$ be a positive integer and fix a non-negative integer $j\le n-1$. Is it true that $$ \sum_{i=j}^{n-1}\binom{n}{i}\binom{i}{j}(-1)^i=\binom{n}{j}(-1)^{n-1} $$ or, equivalently, $$ ...
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93 views

Expressing Factorials with Binomial Coefficients

Expression I have somehow stumbled upon this expression (I believe I have proved it, but that is not important right now), which I have tried to simplify by writing it like something like this (I ...
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3answers
39 views

Binomial coefficients of $p^a-1$ mod p

I need to show that $$\binom{p^a-1}{k}\equiv (-1)^k\mod p$$ where $p$ is a prime, $a$ is a positive integer, and $k<p^a-1$. I think I have done most of the proof, but I'm stuck at the very end. ...
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1answer
78 views

Where does this equation for $n\binom{\binom{n-1}{2}}{m}$ come from?

I'm reading a proof and am having trouble seeing why the following two lines are true: \begin{align*} n\frac{\binom{\binom{n-1}{2}}{m} }{ \binom{\binom{n}{2}}{m}} &= n ...
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342 views

summation of binomial coefficients with squares

What is $$50^2\frac{{n\choose 50}}{{n\choose 49}}+49^2\frac{{n\choose 49}}{{n\choose 48}}...1^2\frac{{n\choose 1}}{{n\choose 0}}$$. i.e. $$\sum_{k=1}^{50} \frac{k^2\binom n k}{\binom n {k-1}}= ?$$ ...
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1answer
44 views

What is $x$ in a polynomial?

I know this sounds like an easy question. But I've never been told this. Suppose we have a polynomial... $$3x^2 + 5x - 9$$ I know $3$, $5$ and $-9$ are coefficients. I also know $2$ (form $3x^2$), ...
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Finite sum with three binomial coefficients

I need to find a closed form expression, if there is one, of the following sum: $$\sum_{j=0}^m{n+1-k\choose j}{k-1\choose m-j}{A+2-k+m-j\choose m-j+2}$$ where all parameters are integers, $~1\leq ...
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217 views

Simplify $\binom nr+ 4\binom n{r+1} + 6\binom n{r+2} + 4\binom n{r+3}+\binom n{r+4}$ [closed]

For $$4\le r \le n,$$ $${n \choose r}+4{n \choose {r+1}}+6{n \choose {r+2}}+4{n \choose {r+3}}+{n \choose {r+4}}$$ equals 1. $${n+4}\choose{r+4}$$ 2.$${n+4}\choose{r}$$ 3.$${n+3}\choose{r-1}$$ ...
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41 views

Binomial expansion of $(x_{1}+x_{2}+…+x_{k})^{n} $ [duplicate]

If we expand $$ (x_{1}+x_{2}+...........+x_{k})^{n} $$ How many terms will be there once we collect terms with equal monomials? What is the sum of all coefficients? I literally have no clue how to ...
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2answers
51 views

how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients?

If you expand $(x_1+x_2+\cdots+x_k)^n$, how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients? I'm kind of lost here. This came up with ...
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1answer
63 views

Looking for a nonrecursive formula for the general derivatives of the quotient of functions

I want to prove that the $k$-th derivative $h^{(k)}(x)$ of the function $h(x)=\frac{1}{1+x^2}$ is zero at $x=0$ for all integer values $k>0$. My only idea was to go the stubborn way applying ...
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1answer
35 views

Find $\sum r\binom{n-r}{2}$

Let $A=\{1,2,3,\cdots,n\}$. If $a_i$ is the minimum element of set $A_i$ where $A_i\subset A$ such that $n(A_i)=3$, find the sum of all $a_i$ for all possible $A_i$ Number of subsets with least ...
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3answers
66 views

How do you deal with fractions in a binomial?

If I have something like this $$\binom{\frac{x}{k}}{\frac{y}{k}}$$ (where there are two fractions in a binomial but they have the same denominator) can I simplify this at all?
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2answers
42 views

What does this binomial sum equal?

I'm trying to evaluate this sum: $$\sum_{k=0}^n {n \choose k}{{2n+1}\choose k}$$ I thought I could work with generating functions of the two binomials. I know $$\sum_k\binom{n}k{}x^k=(1+x)^n$$ is the ...
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0answers
76 views

How to prove this equality about Eulerian numbers?

I want to prove the following equality where $A(k,m)$ is the Eulerian number : $$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$ I previously proved ...