# 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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### GCD of binomial coefficients of the form ($p^n$ choose $k$)

Let $n$ be a positive integer and $p$ be a prime. Find the greatest common factor of $\binom{p^n}{1}, \binom{p^n}{2},...,\binom{p^n}{p^n-1}$. Progress: We know that for any given $n$ and $k$ in ...
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### prove that $(\frac{1}{6})^{4}\cdot\lim_{n\rightarrow\infty}\sum_{i=4}^{n}\binom{i-1}{3}(\frac{5}{6})^{i-4}=1$

I have to prove the following: $(\frac{1}{6})^{4}\cdot\lim_{n\rightarrow\infty}\sum_{i=4}^{n}\binom{i-1}{3}(\frac{5}{6})^{i-4}=1$ any ideas? thanks
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### 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 ...
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### Generating functions and central binomial coefficient

How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4x}}$? More precisely, prove that( for $|x|<\frac{1}{4}$ ): ...
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### relationship between pascal's triangle and number of combinations?

I was able to solve a classic algorithm question, robot paths by using pascal's triangle (PT). This is where a robot starts in the upper left corner and can only go down or right. I kind of reverse ...
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### In how many ways can four persons each throwing dice once sum up to 13?

I am solving it by finding out Coefficient of $x^{13}$ in $(x+x^2+....x^6)^4$ but I cannot get the correct answer. Please provide me the final answer if method I am following is correct.
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### What does ${50}\choose{4}$ mean in statistics?

I have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...
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### Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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### Are there four consecutive binomial coefficients in a row in an arithmetic progression?

Are there four consecutive binomial coefficients in a row in an arithmetic progression? This is suggested by Will Jagy's comment to this question: Find $n$ and $k$ if ...
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### Equation for those level curves?

Related to this question Let $N\in\mathbb{N}^*,\alpha\in\mathbb{R}$. What would be the equation $y=f(x)$ for the curve defined by $\ln\binom{N-y}{x}=\alpha$ That's how they look : TL;DR : What ...
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### Random variable and probability distribution: (5 men and 5 women are ranked)

An exercise in my homework: 5 women and 5 men are ranked based on their results in a test. Assume that all results are different and every of 10! possible orders has the same probability. X (random ...
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### Inequalities involving binomial coefficient when n varies

I am trying to invoque some inequality of the form: $${n+i \choose k} \leq f\left ({n \choose k},n,k \right)$$ where $i$ is small (typically 1, 2 or 3). The tighter the better, however any ...
### when is $\frac{1}{n}\binom{n}{r}$ an integer
So I am considering for which values of n is $a_n =\frac{1}{n}\binom{n}{r}$ an integer for all $1\leq r \leq n-1$. The first thing I did was to check the Pascal Triangle. So I guess n has to be ...