# 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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### Proving a binomial sum with simple result

In deriving a formula I've come up with an expression that I need to prove, specifically: $$(-1)^n = \sum_{j=1}^n (-1)^j \binom{n+1}{j+1} j^n$$ where $n$ is a positive integer. This seems ...
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### Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$

Show that $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$. In the LHS $\binom{n+r-1}{r}$ counts the number of ways of selecting $r$ objects from a set of size $n$ where ...
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### Smallest constant in exponent so that limit of sum is $0$

I am trying to work out the smallest constant $c>0$ so that $$\lim_{n \to \infty} \sum_{a=1}^n \sum_{b=0}^n {n \choose a} {n-a \choose b} \left({a+b \choose a} 2^{-a-b}\right)^{c n/\ln{n}} =0 .$$ ...
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### How prove this $\sum_{k=0}^n \binom{n}{k} \binom{(p-1)n}{k} \binom{pn+k}{k} = \binom{pn}{n}^2$

I think the following equality is true ($p\in \mathbb{N},p\ge 2$): $$\sum_{k=0}^n \binom{n}{k} \binom{(p-1)n}{k} \binom{pn+k}{k} = \binom{pn}{n}^2$$ when $p=2$, then ...
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### How can we find the gcd for elements (binomial coefficient)?

$\gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right)$ i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ...
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### Intuition behind negative combinations

Take $\binom{n}{r}$. It denotes how in how many different ways you can choose $r$ elements from a set of $k$ elements. For case $\binom{4}{3}$ which evaluates to $\frac{4!}{3!(4-3)!}=4$, it perfectly ...
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### Binomial identity

I'd like to get a hint to prove the following identity: $$\tag{1}\sum_{\nu}(-1)^{\nu}\displaystyle \binom{a}{\nu}\binom{n-\nu}{r}=\binom{n-a}{n-r} .$$ The original statement reads "By specialization ...
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### A finite sum involving the binomial coefficients and the harmonic numbers

Wikipedia has a proof of the identity $$H_{n} =\sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$$ http://en.wikipedia.org/wiki/Harmonic_number#Calculation Curiously, there is also the identity ...
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### Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$\frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
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