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 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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46 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
33 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 ...
2
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0answers
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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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2answers
32 views

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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1answer
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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2answers
40 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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0answers
48 views

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, ...
4
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4answers
194 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?
2
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3answers
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
46 views

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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1answer
43 views

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....
2
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4answers
50 views

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
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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
41 views

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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1answer
52 views

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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1answer
62 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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35 views

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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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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30 views

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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1answer
21 views

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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112 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 ...
3
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2answers
112 views

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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1answer
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
54 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
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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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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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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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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: $$(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$$
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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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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 ...
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On a theorem of Hensel

In the paper Binomial coefficients modulo prime powers, Andrew Granville state the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
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Multinomial coefficients modulo a prime

Let $p$ be a prime and let $m \geq 1$. Lucas' theorem implies that the binomial coefficient ${p^m-1 \choose k}$ is not divisible by $p$ for any $0 \leq k \leq p^m-1$. I wonder if something similar ...
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1answer
25 views

Prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$

How to prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$? I tried writing it as an AGP as following: $$\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2} = \textrm{coefficient of } x^...
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42 views

Evaluate $\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $

Evaluate the expression $$\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $$ I'm really stumped about trying to get ...
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1answer
39 views

How can I calculate $\sum_{k=1}^{n-1}\binom{n-1}{n-k}$?

I would like to know if I can calculate a closed expression for $$\sum_{k=1}^{n-1}\binom{n-1}{n-k}$$ This sum is equals to: $$1+(n-1)+(n-1)(n-2)+(n-1)(n-2)(n-3)+\ldots+(n-1)(n-2)/2+(n-1)$$
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26 views

Proof of the diagonalization of the probability matrix of the sum of two binomial distributions

I am analysing a statiscal problem where a vector $X\in\mathbb Z_{\geq 0}^{n+1}$ is probabilistically transformed according to $x_i \mapsto \mathrm{Binomial}(x_i, 1/2+\epsilon/2) + \mathrm{Binomial}(n-...
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1answer
18 views

Is my intuition about this statistics problem sensible?

I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
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2answers
84 views

Verify the following identity algebraically

Verify the following identity algebraically (writing out the binomial coefficients as factorials).$${n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}$$ So far, these are my steps: $$\frac{...
5
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1answer
76 views

Reference request for an identity involving binomial coefficients

The identity is $$\sum_{i\ge 0}(-1)^i \binom{\frac{s^k + s^{-k} - 10}{4}}{i}\binom{\frac{s^k + s^{-k} - 10}{4}+4}{\frac{gs^k + 2g^{-1}s^{-k} - 4}{8}-i}= 0$$ where $k\gt 1$ , $s = 3+2\sqrt{2}$ ...
0
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0answers
51 views

Sum of Two Double Sums

Suppose $ r^{2}=4ab $. Show that over the complex number field: $$ \left( \sum_{k=0}^{l}{\sum_{m=0}^{l-k}{\binom{l}{k}\binom{l-k}{m}a^{l-k-m}b^{m}(-1)^{m}r^{k-1}i^{k-1} \left( \frac{2a-(-1)^{k}\left(...
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1answer
45 views

Find the sum of the series .

The general term of the series is $\sum_{r=0}^{100}\binom{500}{r}\binom{500-r}{400}2^{100-r}$ What I tried to think that this series was an expansion for a series inside a series but the thing that ...