# Tagged Questions

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### Bound the expression with Stirling approx.

I already compute 2 expressions of one problem, but they are so complicated as below. My prof requests me to reduce them or upper bound them by Stirling approximation but I have not succeeded. 1/ ...
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### Primes Not Dividing $\binom{2n}{n}$

Let $n \geq 3$, show ${2n \choose n}$ is not divisible by $p$ for all primes $\frac{2n}{3} <p\leq n$ Note: This fact along with other facts about ${2n \choose n}$ are used in a proof of Bertrand's ...
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### Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

How can I prove the following, where $p$ is a prime and $x$ a positive integer? $$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$ I'm not sure if it is actually true, but I tested ...
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### Fractional Binomial Coefficients

I recently examined the binomial coefficient $\binom{\frac{1}{2}}{k}$ and found that the denominator was always a power of two. The same is true of $\binom{\frac{1}{3}}{k}$, where the denominator is ...
### Does $k$-th power of $p$ divide ${}_n\!C_r$ if the previous divides $n$?
Does $p^k$ divide ${}_n\!C_r$ for all integer r if $p^k|n$ where $0\leq r \leq n$ and $p$ is prime?