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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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
111 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\}} ...
2
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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 ...
0
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0answers
43 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
191 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 $n=5$, ...
2
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1answer
45 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
41 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 ...
2
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4answers
47 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
54 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
50 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
35 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 ...
2
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1answer
49 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 ...
2
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2answers
16 views

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 ...
0
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1answer
18 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
60 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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0answers
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, ...
4
votes
2answers
110 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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1answer
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 ...
4
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4answers
110 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
106 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 $$\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$$
2
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1answer
136 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 $101=102-1$, ...
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1answer
24 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 ...
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1answer
53 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?
2
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1answer
55 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 ...
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1answer
32 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
152 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
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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: $$(n+1)\binom{n}{k}=(k+1)\binom{n+1}{k+1}$$
4
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2answers
39 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 $n.$ ...
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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 ...
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0answers
23 views

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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16 views

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 ...
3
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1answer
24 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 } ...
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2answers
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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0answers
25 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) + ...
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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: ...
5
votes
1answer
75 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
votes
0answers
50 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( ...
0
votes
1answer
44 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 ...
2
votes
0answers
36 views

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 ...
1
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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 ...
2
votes
0answers
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 ...