binomial-coefficients's questions - Chinese 1answer

4.247 binomial-coefficients questions.

We can be sure that for $a>1$ $$\sum\limits_{k=1}^{\infty}(-k)^na^{-k}+\sum\limits_{k=0}^{b}k^na^k=a^b\sum\limits_{m=0}^{n}\binom{n}{m}\left[b(a-1)\right]^{n-m}\sum\limits_{k=0}^{\infty}(-k)^ma^{-k}...

I'm trying to reproduce Excel's COMBIN function in C#. The number of combinations is as follows, where number = n and number_chosen = k: $${n \choose k} = \frac{n!}{k! (n-k)!}.$$ I can't use this ...

Define the $q$-binomial (Gaussian) coefficient ${n+m\brack n}_q$ as the generating function for integer partitions (whose Ferrers diagrams are) fitting into a rectangle $n\times m$, i.e., for the set $...

Is it possible to calculate $\sum_{k=0}^{20}(-1)^k\binom{k+2}2$ without calculating each term separately? The original question was find the number of solutions to $2x+y+z=20$ which I calculated to ...

Show that $\binom{2n}{n}\geq2^n$ for all $n\in\mathbb{N}_0$. First, observe that $\binom{2n}{n} = \frac{(2n)!}{n!(2n-n)!}=\frac{(2n)!}{(n!)^2}=\frac{1\cdot 2\cdot 3\cdot...\cdot 2n-1\cdot 2n}{(n!)^2}=...

Consider the following matrix where the rows are coefficients of sums of powers of $n$: $$\left( \begin{array}{cccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ ...

While I was playing with Wolfram Alpha online calculator I wondered that I know how to calculate with the help of this tool and my knowledges the first cases for integers $n\geq 1$ of this type of ...

I am figuring out a possible pattern which somehow follows the pascal triangle. Suppose we have the following: $${{a}\choose{b}}+{{c}\choose{d}}+{{e}\choose{f}}+{{g}\choose{h}}={{8}\choose{1}}$$ I ...

Let $s_2(n)$ denote the sum of digits of $n$ in base-2 (OEIS sequence A000120), and $t_n=(-1)^{s_2(n)}$. Note that $t_n$ is the signed Thue–Morse sequence (OEIS sequence A106400), satisfying the ...

How to find the numerically greatest term in the expansion of $(3x+5y)^{12}$ when $x=\frac12,y=\frac43$? My attempt $$(3x+5y)^{12}=\left(3x\left(1+\frac{5y}{3x}\right)\right)^{12}$$ $$=3^{12}x^{12}\...

Let $q=p^r,$ where $p\in\mathbb{P}$ is a prime and $r\in\mathbb{N}\setminus\{0\}$ is a natural number (non-zero). How to prove that for each $i\in\{1,2,\ldots,q-1\}$ the binomial coefficient $\binom{q}...

I am trying to find limit of the following function: $$\lim_{n\rightarrow \infty}\frac{\sum\limits_{x = 0}^{n}\binom{n}{x}\left[1 + \mathrm{e}^{-\frac{x+1}{n+1}}\right]^{n + 1}}{\sum\limits_{x = 0}^{...

Honestly, I have no idea if I put the correct tag on this question, and I don't even know where to begin to solve an equation like this: $$ f(d,n)=\sum_{i=1}^n\binom{d}{i}. $$ Could someone explain ...

‘Show how the binomial expansion can be used to work out $268^2 - 232^2$ without a calculator.’ Also to work out 469 * 548 + 469 * 17 without a calculator. I understand the process of binomial ...

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...

Compute the sum $\sum_{k=0}^{n}(-1)^k k^n\binom{n}{k} $ I've seen a solution along the following lines here, page 3: Consider $(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k$. ($\star$) We prove by ...

I just realised the following identity: $$\sum_{i=0}^n {2n+1\choose i}=\sum_{i=0}^{2n}{2n\choose i}.$$ Is that correct? Has this been already proved?

Evaluate the following sum involving binomial coefficients, $$ \sum_{k=0}^n \binom{n}{k}\frac{k!}{(n+1+k)!} $$ when $n$ is a nonnegative integer.

Problem: The series$\sum\limits_{n=1}^{\infty}(\frac{(2n-1)!!}{(2n)!!})^p$ converges if and only if $p>2$ My question is: can anyone solve the problem by using binomial coefficients or Stirling's ...

show that if $xy=ax+by$ then $$ x^{n}y^{n}=\sum_{k=1}^{n}\binom{2n-1-k}{n-1}(a^{n}b^{n-k}x^{k}+a^{n-k}b^{n}y^{k})\quad\quad for\:all\quad n>0.$$ Finding a similar formula for the more general ...

I am trying to find the closed form for this binomial summation term. $$ \sum_{j=1}^{m}(-1)^{j+1}\binom{r}{j}\sum_{k=1}^{n}\binom{-j+rk+s}{m-j}, \quad\quad\quad integers\quad m,n\geq0.$$ Need ...

I'm trying to find an asymptotic expansion for the following sum: $$\sum_{k=2}^{n}\frac{n!}{k\left(n-k\right)!}=\sum_{k=2}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)\left(k-1\right)!$$ for ...

Given the polynomial ${1\over8}((1+z)^9 + 3(1-z)^4(1+z)^5 + (1-z)^6(1+z)^3)$ (which is the weight enumerator of a code) how do I find out the coefficient of $z^2$? The solution given is ${1 \over ...

I have the following equation: $${{x}\choose{3}}=10$$ I want to solve it for $x$ so I wrote: $${{x}\choose{3}}=\frac{x!}{3!(x-3)!}=10$$ which follows: $$\frac{x!}{(x-3)!}=60$$ I wonder how to ...

We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(1)=1$. You can see this ...

So I have a rectangle, that has points A and C on the y-axis, and A at (0,0). Then it is stretched along the y-axis. I have all new coordinates of the stretched rectangel. How to calculate the ...

Review a preprint named Johann Faulhaber and Sums of Powers by Donald E. Knuth, on page 10 are shown sums of $n$ to odd powers, for example, $$\sum n^1=\binom{n+1}{2},$$ $$\sum n^3=6\binom{n+2}{4}+\...

Suppose I have this expression: $$\prod_{n=1}^{N} (1- x^n)$$ In the expansion of the product series how to find the general formula for finding the coefficient of $x^n$ ?

Let $p$ in $(0,1)$. How to compute $$\sum_{k=1}^{\infty} 2k{2k\choose k+1} p^{k-1} (1-p)^{k+1}\ ?$$ Personal tries: I've tried to use generating functions, but I can't deal with this $2 k$ in front ...

How can we prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$ (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.

I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {...

I have a rather standard problem. Suppose that 10 unfair coins are tossed. What is the probability of obtaining 4 successes (H) where the probability of success (H) is 6/7 and failure (T) is 1/7? I'm ...

My background is not mathematics and I need to implement (in C++) the derivative of a binomial, with wxMaxima and wolfram.alpha as a helper. So far, the binomial can be written as: $$\binom x n = \...

In birthday problem say total number of people n < 365, then probability of all person having distinct birthday is given by, $$\frac{\text{total no. of ways of selecting $n$ numbers from $365$ ...

$${{n}\brace l+m}\dbinom{l+m}{l}=\sum_{k \in \mathbb{Z}}{{k}\brace l}{{n-k}\brace m}\dbinom{n}{k}$$ LHS: gives the ways ways to partition $[n]$ into $l+m$ blocks with $l$ blocks (lets say) underlined....

Edit: I've tried to make this question more clear. I know how algebra and calculus work, and can solve problems like these, where the point of the problem is to just simplify a term into its simplest ...

A teacher gave this as a homework question, and I have tried but haven't been able to arrive at a solution. $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ Could someone prove it, or at least ...

Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$ ? I know that $\sum_{k=0}^{n}\binom{n}{k}= 2^{n}$ and so, $\sum_{k=1}^{n}\binom{n}{k}= 2^{n}-1$ but how to deal with $k$ ?

I'm still learning about this so any clarification would be very helpful What is the purpose of quadratics equations? How did Mathematician come up with it? How did they figure out the formula for ...

In the middle of a proof, I have had to analyze the asymptototic behavior of $$ \mathbb{E}\left[\frac{1}{(1+X)^2}\right] = \frac{1}{2^n}\sum_{k=0}^n \binom{n}{k}\frac{1}{(1+k)^2}\tag{1} $$ where $X$ ...

Find the coefficient of $x^{17}$ in the expansion of $(3x^7 + 2x^5 -1)^{20}$ I'm stuck in handling this question as I do know how to solve it when it has 2 terms. But now it has 3. I have no idea ...

We know that the binomial theorem and expansion extends to powers which are non-integers. For integer powers the expansion can be proven easily as the expansion is finite. However what is the proof ...

I would like to know the difference between "permutations with repetition" and "ways to choose $k$ elements from a set of $n$ elements if repetitions are allowed". For instance: In a set $S$ of $k$ ...

For instance, will $(x+y)^{-n}$ be equal to $1/((x+y)^n)$? I know this is a silly doubt, but since some $x^{-a} = 1/x^a$, I just wanted to know if this is applicable here also.

I come across the following binomial sum when studying the average-case time complexity of an algorithm: $$\sum_{i = 1}^{n-k+1} i \binom{n-i}{k-1}$$ How to evaluate this sum?

Good afternoon. I'm looking for a proof, or a counterexample that, given $n,k,N\in\mathbb{Z}$, with $n>k>0$, $N\ge2$, $$(N+1)|\binom{nN}{kN}$$ Just using the definition, $$\binom{nN}{kN}=\...

a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +...

Let $\beta_1$ , $\beta_2$ and $S$ real numbers, and $N$ and $M$ natural numbers. Can we simplify this formula. $$ \sum_{n=0}^{N-1}\sum_{m=0}^{NM-1}(-1)^{n+m}\binom{N-1}{n}\binom{NM-1}{m} \beta_{1}\...

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