# 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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### Asymptotics of ${2^n \choose n}$?

How can one compute the asymptotics of ${2^n \choose n}$? I know it is bounded below and above by $\left(\frac{2^{n}}{n}\right)^n$ and $\left(\frac{2^{n}e}{n}\right)^n$. If I plug in Stirling's ...
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### Coefficient of $x^7$ in binomial expansion of $(1/6-3x)^{17}$

How would I determine the coefficient of $x^7$ in the expansion of $(1/6-3x)^{17}$ and show the answer as a fraction?
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### General Leibniz rule for triple products

I have a question regarding the General Leibniz rule which is the rule for the $n^{th}$ derivative of a product and reads: $$(f g)^{(n)}=\sum_{k=0}^{n} {n \choose k} \,f^{(k)} g^{(n-k)}.$$ However,...
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### Proving Bernoulli's Inequality for $h<0$

I'm answering question 19 of chapter two of Spivak's Calculus and I can't seem to think of a way of doing it. I don't want to look up the answer so I thought I'd ask for a hint as to the general ...
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### Binomial coefficients identity (sum of the powers of the natural numbers)

I've found exercise with binomial coefficients in Kostrikin's book. Proof that $\sum_{i=1}^n{{r+1}\choose{i}}\left(1^i+2^i+\dots+n^i\right)=(n+1)^{r+1}-(n+1)$ I was trying to check that for ...
129 views