# 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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### Which one is greater $600!$ or $300^{600}$

Which one is greater $600!$ or $300^{600}$ $\bf{My\; Try::}$ I have used Stirling Approximation. For large $n>2\;,$ We can write $\displaystyle n! \approx \left(\frac{n}{e}\right)^n\sqrt{2\pi n}$...
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### How to find the value of sum of even binomial coefficient? [duplicate]

I want to find the value of $\sum_{i=0}^k {n \choose 2i}$ where $2k\,\le\,n$. Is there any short formula to find the answer.
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### Is there a rigorous proof of this combinatorial identity?

Theorem: For any pair of positive integers $n$ and $k$, the number of $k$-tuples of positive integers whose sum is $n$ is equal to the number of $(k − 1)$-element subsets of a set with $n − 1$ ...
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### Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...
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### A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
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### identity on Pascal's triangle modulo 2

Consider Pascal's triangle with entries modulo $2$, and let $(k,l)$ denote the $l$-th entry in the $k$-th row by $(k,l)$. Show that, for all $n \in \mathbb{N}$, each entry of the triangle with ...
### Prove that $\sum_{j=0}^{n}H_j{n\choose j}^2={2n\choose n}\left(2H_n-H_{2n}\right)$
Let $H_n$ the $n$th Harmonic numbers and $H_0=0.$ Prove that $$\sum_{j=0}^{n}H_j{n\choose j}^2={2n\choose n}\left(2H_n-H_{2n}\right)$$ I encounter this problem since 2012 and have verify ...