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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Double counting proof of binomial problem

The assignment is to prove the following assertion using the method of double counting and explaining which pairs were counted. $$\dbinom{n+1}{k+1} = \sum_{i = k}^{n} \dbinom{i}{k}$$ Left side is ...
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65 views

Finding the coefficient of expansion

Question: Find the coefficient of $x^{11}$ in the expansion of:$$(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$$ The traditional way of doing this, as far as I know, is to first find the coefficient of each term ...
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1answer
31 views

Of Balls in Bins in Different Sections with Caps

Problem: There are $19$ bins: $7, 5, 7$ in the left, centre and right sections respectively. There are $8$ balls, some or all of which are to be put into these bins with the following ...
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1answer
34 views

Combinatorial identity / expected distance of random walk

I am struggling to verify the following identity. $$\binom{2m}{m} \frac{m}{2} = \sum_{j=1}^m j \binom{2m}{m+j}$$ I've tried induction, but I run into issues inside the sum. I can't see a ...
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37 views

Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
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Computing the coefficient of $x^n$ in the following expansion

The coefficient of $x^{-n}$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is $a.)$ $(-3)^n - (2)^{\frac{1}{2}n -1} $ $b.)$ $2^n + 1 $ $c.)$ $ 3(2)^{\frac{1}{2}n - 1} - 2(3)^n $ $d.)$ None of the ...
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40 views

Closed form of a finite sum with binomial coefficients

In general, has $\sum_{k=a}^{b} \binom{n}{k}$ a closed form (with $0\le a\le b\le n$)?
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55 views

Check whether or not a triangular number is triangular is the square-sum of two other consecutive triangular numbers

I'm trying to write a program that would tell me whether or not a triangular number, a number of the form $\frac{(n)(n+1)}{2}$ is the sum of the squares of two other consecutive triangular numbers. It ...
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27 views

Coefficient of Multinomial kind of expression

How do I find the Multinomial coefficient of expression. For example $(x+y+z+w+6)^8$ let say I want the coefficient of xyzw. I know the answer in the simple case of $(x+3)^5$ , for $x^2$ it will ...
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Is there a name for this combinatorial identity?

I found this identity in a textbook that I own but they did not name the identity and I had some trouble finding it online. Does anyone know the name of the identity and if I can find a resource about ...
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182 views

Proving a binomial coefficient identity [duplicate]

I'm having some trouble with the following proof: $$\sum^k_{a=0} {{n}\choose{a}}{{m}\choose{k-a}} = {{n+m}\choose{k}}$$ I'm trying to prove this to learn a couple of things about the Pascal's ...
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How in this world can I simplify this $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????

I have a problem, obviously. I am doing some maths and now I have to simplify this: $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????. But I just don´t know how ???? I´ve started simplifying by ...
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1answer
49 views

Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
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1answer
37 views

Binomial Expansion - Finding the term independent of n.

The coefficient of $x^2$ in the expansion of $\left(1 + \frac x5\right)^n$, where $n$ is a positive integer, is $\frac 35$ . $(i)$ Find the value of $n$. $(ii)$ Using this value of $n$, find ...
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Why is $\sum\limits_{k:|2k/l-m/l|\geq\epsilon/2}\frac{\binom{m}{k}\binom{2l-m}{l-k}}{\binom{2l}{l}}\geq 2e^{-\epsilon^2l/8}$

This is stated in an article on the uniform convergence of probabilities of events to their relative frequencies. The idea behind the question is that I have a measurement on a sample of size $2l$ ...
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Proof of the binomial identity $\displaystyle\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}$

Trying to prove some uncorrelated things, I came across the following identity: $$\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}, $$ where $\delta_{...
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Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\displaystyle \sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}$ for $1 \leq \alpha \leq p$? EDIT: This is a result that I derived ...
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Prove by induction: the coefficients of (a+b) to the power of n are the same if turned into a number as 11 to the power of n

Proof by induction that the coefficients of $(a+b)^n$ in order, if place as a number, the first coefficient being having the biggest place value, and each number lowers in place value, are equal to ...
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2answers
50 views

Value of the the sum of reciprocals of combinators

Evaluate $$\sum_{n=2009}^{\infty} \frac{1}{ \binom{n}{2009}}$$ I tried making the $r^{th}$ term as a difference of 2 terms, but that didn't work out. Do we need to integrate or something?
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112 views

How prove this algebraically? [closed]

Let $a,b,x,y$ be nonnegative integers. By way of using generating functions, prove that $$ \begin{pmatrix} x+b \\ a \end{pmatrix} \begin{pmatrix} y+a \\ b\\ \end{pmatrix} = \sum_{i=0}^{\min\{a,b\}} ...
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41 views

Sitting n families around a circular table with a condition

How many ways are there for sitting n families around a circular table. Each family is a mother a father and a child. Condition: The mother and father of each family should be sitting next to each ...
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An Identity with Binomials and Harmonic Numbers

Let $m,n,p$ positive integers with $m\geq n$ and $H_m=1+1/2+1/3+\cdots+1/m$ the $m-$ith Harmonic Number with $H_0:=0$. Show that for the values of $m,p,n$ for which the denominators do not vanish, ...
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4answers
202 views

How do I show that $\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$? [duplicate]

$$\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$$ I know how to "prove" it by interpretation (using the definition of binomial coefficients), but how do I actually prove it?
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81 views

Surprising Summation (4): $\frac 12 \sum_{i=1}^n (n+1-i)(n+i)=\sum_{i=1}^n i^2$

Show that $$\frac 12 \sum_{i=1}^n (n+1-i)(n+i)=\sum_{i=1}^n i^2$$ without expanding the summation to its closed form, i.e. $\dfrac 16n(n+1)(2n+1)$ or equivalent. e.g.for $n=5$, $$\frac12\bigg[5(...
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1answer
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showing that ${kp^2\choose jp^2} \equiv {k\choose j}$ modulo $p$

Given $1\le k \le p-1$ and $1\le j \le k$, show that ${kp^2\choose jp^2} \equiv {k\choose j}$ modulo $p$ where $p$ is some prime integer. Could I receive some hints? I tried writing the expressions ...
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Inverse Euler theorem to calculate $\binom{n}{r}$

How can we use Inverse Euler theorem or properties to calculate the binominal coefficients or say $\binom{n}{r}$? What is the algorithm for this ? An example for the same will be greatly appreciated....
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Coefficients and Expansions

So I was just hoping for a look over my work to check if what I am doing is right because I'm not so sure: Find the coefficient of: f$$ x^6$$ with the equation $$(3x-\frac{(1)}{x^2})^{12}$$ I have: $...
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2answers
55 views

Binomial coefficient definition

Why is the definition of the binomial coefficient $${{m}\choose {r}}=\frac{m(m-1)\cdots(m-r+1)}{r!}$$ I'm not sure where the last term in the numerator came about. Why should there be a $+1$? ...
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1answer
51 views

Problem with binomial coefficients?

I am trying to find the sum of $$\sum_{x=0}^{n-2}\left (\frac{1}{x+1}{2x \choose x} \cdot \frac{1}{n-x-1}{2n-2x-4 \choose n-x-2}\right)\;.$$ I am told the answer is $$\frac{1}{n}{2n-2 \choose n-1}$$ ...
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1answer
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Algorithm for calculating multiset permutations

I have this algorithm to calculate multiset combinations: $$\mathcal P(k; m_1, m_2, \ldots, m_n) = \Sigma \binom{c(i_1)}{\lambda_1}\ \binom{c(i_2)-\lambda_1}{\lambda_2} \cdots \binom{c(i_s)-\lambda_1-...
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what is the probability of bin 1 was chosen

Bin 1 contains 20 parts, 5 are defective. Bin 2 contains 15 parts, 4 are defective. One of these Two Bins is chosen at random and 3 parts are randomly selected from the bin chosen. if 2 of the 3 parts ...
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Probability of odd-man when $n$ people tossing a coin with probability $p$ of getting head

Οdd-man means a person gets a different result from all other people. Probability of getting a head is $p$. Number of coins knowing the number of people is $n$. So number of possible ways to get ...
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1answer
20 views

number of subsets of a set with even sum using combinatorics or binomial

Let S={a1,a2,a3.......aN}.There are 2^N subsets of this set so if we don't consider the empty set we are left with 2^N-1.We do need to consider cases where it number of odd numbers may be zero and ...
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64 views

Value of sum of binomials: $P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ [duplicate]

$P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ I can calculate the value of this equation manually, but there any direct formula for calculating the value of ...
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prime numbers - need a help

Helow, There is a question about prime numbers. Supposed that I already answer the first section. I try to answer the second section, but if n $\neq$ $2^{k}$ (for some k from the natural numbers, ...
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2answers
113 views

Solving ${98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+…+68{31 \choose 30}={100 \choose q}$ for $q$

${98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+...+68{31 \choose 30}={100 \choose q}$ Find the value of $q$? Could someone give me hint as how to solve this question?
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Binomial inequality problem ${k+n-1 \choose k}\times{k+n+1 \choose k} \leq{k+n \choose k}^2$

Can anyone help we with this problem: Let $a_n={k+n \choose k} $ Prove that $a_{k-1}a_{k+1}\leq a_k^2 $($\forall k$) My first idea was using mathematical induction to proof that for every k element of ...
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How find the value of $(a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2$ using $(1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n$?

Q) $(1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n$ then $(a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2$ is equals to 1. 12. 0 (zero)3. $2^{n-1}$4. $2^n$ Answer: (4) well this time i am rocked by this ...
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114 views

Can anyone give a combinatorial proof of the identity ${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+…+(n-m+1){m \choose m}={n+2 \choose m+2}$

Can anyone give a combinatorial proof of the identity $${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+\ldots+(n+1-m){m \choose m}={n+2 \choose m+2}$$ I am finding difficult as $n$ is varying ...
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sum of series $\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$ [closed]

Does there exist an explicit formula for the sum of the series $$\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$$
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155 views

Sum involving binomial coefficients.

Prove that $${^{404}\mathrm C_4}-{^4\mathrm C_1}\cdot{^{303}\mathrm C_4}+{^4\mathrm C_2}\cdot{^{202}\mathrm C_4}-{^4\mathrm C_3}\cdot{^{101}\mathrm C_4} =(101)^4$$ I tried writing $101=102-1$, but ...
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1answer
25 views

Why does $\sum_{k\geq0}\binom{-r}{k}p^r(p-1)^k=p^r(1+p-1)^{-r}$?

For any positive real number $r$, it is clear that $\binom{-r}{k}(-1)^k\geq0$ for all positive integer $k$. The general binomial theorem then implies $$\sum_{k\geq0}\binom{-r}{k}p^r(p-1)^k=p^r(1+p-1)^...
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1answer
57 views

Is Binomial:Gamma ever an integer?

We consider: $$\dfrac{\Gamma(n)}{\Gamma(k)\Gamma(n-k)}\quad\quad[1]$$ for $n,k\in\mathbb{R}$. Is $[1]$ ever an integer, except for the obvious?
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1answer
56 views

The value of $\mathop{\sum\sum}_{0\leq i< j\leq n}(-1)^{i-j+1}\binom{n}{i}\binom{n}{j}$

The value of $$\displaystyle\mathop{\sum\sum}_{0\leq i< j\leq n}(-1)^{i-j+1}\binom{n}{i}\binom{n}{j} = $$ $\bf{My\; Try::}$ Let $$S=\mathop{\sum\sum}_{0\leq i<j\leq n}(-1)^{i-j+1}\binom{n}{i}\...
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1answer
34 views

Missing notation in one of the test questions

I am looking at STEP (Cambridge produced test for maths) questions and have stumbled upon this question. Does anyone know what notation is missing here in the section i). Please do not provide me with ...
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2answers
153 views

Evaluating a “binomial-like” sum

I suspect there is a way to do the following sum by hand, but I'm having some trouble: $$\sum_{x=0}^{n} x^{2} {n \choose x} p^{x}(1-p)^{n-x}$$ There are a couple questions like this, but for general $...
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3answers
41 views

Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively.

Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively.Find $(a-b).$ I tried to factorize $(1+x+2x^2+3x^3)$ and $(1+x+2x^2+3x^3+4x^4)$ into ...
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2answers
51 views

How to prove that $(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$? [closed]

How to prove that for the integers $k,n$ where $k \leq n$ the following holds: $$(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$$
4
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2answers
40 views

Let $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If $f(n)=2047,$ then find the value of $n.$

Let $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If $f(n)=2047,$ then find the value of $n.$ $f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}=\sum_{k=0}^{n}\binom{k}{0}+\sum_{k=1}^{n}\binom{k}{1}+\...
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3answers
71 views

Prove algebraically that ${n\choose k}=\frac{n(n-1)…(n-k+1)}{1\cdot 2\cdot …\cdot k}$

From the definition of binomial coefficient, $${n\choose k}=\frac{n!}{k!(n-k)!}\Rightarrow \frac{n!}{k!(n-k)!}=\frac{n(n-1)...(n-k+1)}{k!}$$ $$\Rightarrow \frac{n!}{(n-k)!}=n(n-1)...(n-k+1)$$ Could ...