# Tagged Questions

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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### Proof of $\sum_{n=1}^\infty \frac{1}{n^4 \binom{2n}{n}}=\frac{17\pi^4}{3240}$

Recently, I was able to prove that $$\sum_{n=1}^\infty \frac{1}{n \binom{2n}{n}}= \frac{\pi}{3\sqrt{3}}$$ $$\sum_{n=1}^\infty \frac{1}{n^2 \binom{2n}{n}}= \frac{\pi^2}{18}$$ But does anybody know ...
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### What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial coefficient ...
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### Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
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### Proving $\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$

Prove that $$\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$$ by computing the coefficient of $z^M$ in the identity $$(1 + z + z^2 + \cdots ) \cdot \frac{1}{(1-z)^{k+1}} = \frac1{(1-z)^{k+2}}.$$ I ...
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### Combinatorial proof for $\sum_{i=r}^{n}(2i-r)\binom{i-1}{r-1}^2=r\binom{n}{r}^2$

I cfound the following identity, but I'm having trouble finding a combinatorial interpretation. Can someone help me? $$\sum_{i=r}^{n}(2i-r)\binom{i-1}{r-1}^2=r\binom{n}{r}^2$$
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### How can I calculate $\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}$

I tried Google and various ways, including walking the list questions in the chronological order as far. How can I show $$\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}=2^{2n}$$ ...
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### binomial calculation method

I want solve this probability: For $p= 0.4$ $q=0.8$ $n= 20$ $1-P(5<x<11)$ = $1-\sum_{k=6}^{10} \binom{20}{k}(0.4)^k(0.6)^{20-k}. Wolfram Alpha -> = 0,2531$ Is calculation method ...
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### Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
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### Formula for $\sum_{i\geq 0} i{n \choose 2i}$?

So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone ...
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### Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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### # of bit strings of length n (even>2), with n/2-1 zeros and n/2+1 ones, zero followed by one

case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one Answer: 3 case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, ...
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### Prime Factors in Pascal's Triangle

The question about reversing n choose k made me look a little further into Pascal's triangle, but my curiosity is not satiated. I am now curious of the following: Given $n > k > 1$, show ...
I got this question and I am little bit confused, whether the question is correct or not. $n\choose0$-$n\choose1$+$n\choose2$-$n\choose3$+.........+$(-1)^r$$n\choose r=28 Now we are ... 2answers 64 views ### Understanding steps in a proof of \sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1} So, the task is to prove that:$$\sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1}$$I tried different methods but none led me to the solution. I looked up the solution and I can't even ... 0answers 31 views ### How to fast compute the coefficient of x^{2^{n-1}} in polynomial [\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]]^{2^{n-r+1}}? How to fast compute the coefficient of x^{2^{n-1}} in polynomial$$\left(\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]\right)^{2^{n-r+1}}$$or compute the coefficient of any term x^k for k\geq 0? ... 3answers 68 views ### How do you evaluate this sum of multiplied binomial coefficients: \sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} ? We have to find the value of x+y in:$$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$My approach: I figured that the required summation is nothing but the coefficient of x^3 is the ... 2answers 91 views ### Sum of binomial coefficients \sum_{k=n}^{r}\binom{k}{n} Is is possible to find the sum of a binomial coefficient series like: \sum_{k=n}^{r}\dbinom{k}{n}? Just a random thought. 2answers 129 views ### A combinatorial identity \sum_{i=0}^k \binom ni \binom{-n}{k-i} =0 Can anyone prove the following identity for me? \sum_{i=0}^k \begin{pmatrix} n\\ i \end{pmatrix} \begin{pmatrix} -n\\ k-i \end{pmatrix}=0 for any positive integers n,k. I'm pretty sure this is ... 2answers 51 views ### Binomial Coefficient Identity Involving Summation Prove that$$\sum_{j=0}^n (-1)^j \binom{n+j-1}{j}\binom{N+n}{n-j} = \binom{N}{n} $$I tried to prove this via binomial expansions of (1-x)^N (1+x)^{-m}, and equating the coefficients of x, ... 6answers 3k views ### Elementary central binomial coefficient estimates How to prove that \quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad for all n > 1 ? Does anyone know any better elementary estimates? 1answer 57 views ### A sum of squared binomial coefficients I've been wondering how to work out the compact form of the following.$$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}$$3answers 38 views ### Multiple objects, Number of combinations A bag contains colored balls: 8 red balls 4 white balls 4 blue balls 4 green balls 2 purple balls 2 orange balls 1 yellow ball 1 black ball Total of 26 balls. I'd like to determine the number of ... 2answers 52 views ### Triangular numbers and pascal's trangle The following are the triangular numbers. rank = 1 2 3 4 5 6 term = 1 3 6 10 15 21 A rule for triangular numbers is: ... 2answers 92 views ### What is the probability that when flipping a fair coin ten heads will be thrown in a row I just cant figure this out.. Should I be using binomial distribution? Chance of getting ten tail long series in 100 coin throws 1answer 649 views ### Calculating the Shapley value in a weighted voting game. Given a special case of WVG (Weighted Voting Game) of a 1s and b 2s and a quota q, [q:1,1,1,1..1,2,2,..2] . I need help with calculating the Shapley value of a player with a weight of 2 and a ... 3answers 110 views ### Finding the coefficient of x^7 in the expansion (1 + x)^{23} By definition, the Binomial Theorem states:$$(x+y)^n = {n\choose 0}x^n + {n\choose 1}x^{n-1}y + {n\choose 2}x^{n-2}y^2 + \cdots + {n \choose {n-1}}xy^{n-1} + {n \choose n}y^n$$For any$x,y\in\...
I want to find the following sum mod $p$ (prime number): Let $i\geq \frac{p-1}{2}$, $\sum_{k=i}^{p-1} \binom{k}{i}\binom{k}{p-i-1} \pmod{p}$ OK, I succeeded in simplyfying this argument to the ...