# 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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### Ways to think about the binomial coefficient

Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial ...
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### Sum involving integer compositions and binomial coefficients

I came across an interesting identity involving binomial coefficients. I'm not sure if I'm looking at the identity the wrong way but I am not aware if this identity is known and if there is an (easy) ...
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### For any given $k$, show that an integer $n$ can be represented as: $n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$

For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$. My ...
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### Does the property ${n\choose r}={n\choose n-r}$ have a name?

Due to the relation between Pascal's Triangle and the choose function in probability theory, we can deduce that $${n\choose r}={n\choose n-r}$$ because Pascal's Triangle is symmetric. This can also be ...
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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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### Binomial random variable/z-score question?

I was given the problem: In a restaurant called ”Allegory”, on average 1 in 10 people order a bottle of white wine. Out of a sample of 50 people 11 chose a bottle of white wine. Has this wine become ...
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### Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

For a given k, ${n\choose k}$ is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding base p digit of k (consider the p-ary notation for n ...
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### Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
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### How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
### Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$
We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...