# Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### Factorials dividing sums of factorials

For the case of a factorial dividing another factorial, one has that $m! \mid n!$ if and only if $m \le n$, for any two positive integers $m$ and $n$ (yes, they must be strictly positive, because ...
1 vote
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### When does a line equal the Gamma function?

How would you solve for $x$ in the following equation: $$x = \left( x - 1 \right) ! = \int_0^\infty t^{x-1} e^{-t} dt$$ If we are only concerned about integers, then clearly, the only solution is $1$...
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### Proving $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer using number theory [duplicate]

This is an intermediate result for an AMC problem: 2019 AMC10A Problem 25. The solution presents two ways to determine that $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer. I understand the combinatorial ...
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### How would you go about finding the exponent for any given x that most closely matches or surpasses factorial growth?

While analyzing the factorial function and comparing it to basic exponentiation, I couldn't help but notice the obvious fact that exponentiation can eventually overtake factorialization if the ...
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### What is the asymptotic version of the solid angle formula in $d$ dimensions?

It is well known that the solid angle in an euclidian space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$, where $n = 1, 2, 3, \dots, \infty$) is given by these formulae: \begin{align}\tag{1} \Omega_{...
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### Any good analytical approximation for the inverse factorial moment of Poisson random variable?

I have the following question: Consider a random variable $M \sim \mathrm{Poisson} (\lambda)$. I would like to evaluate the following expectation: \begin{align*} \mathbb{E} \left[ 1 / \Gamma (M + a) \...
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### Closed form solution for indexing combinations from n choose r

I have a list of combinations resulting from n choose r where order doesn't matter and without repeats, and the list is ordered based on the first choice. To make this concrete, in my specific case, ...
1 vote
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### Are there infinite triplets that satisfy a! = b!⋅c! where a ≠ b+1? [duplicate]

I'm not a mathematician. I was recently watching a YT math question on what is 10! ÷ 6! = x!. It got me thinking, are there infinite triplets (a, b, c) that satisfy a!= b!⋅c!? I wrote a short python ...
1 vote
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### A number-theoretic proof that $(n^2)!/(n!)^{n+1}$ is an integer [closed]

I have seen a number of combinatorial proofs for this statement. For instance, Quotient of factorials However, I am wondering whether there is a purely number-theoretic proof.
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### But why does any $\Gamma(z)\Gamma(1-z)\gets\pi\csc(\pi z)$?

So I was looking through my questions for a sense of nostalgia when I came across this question of mine asking on how to evaluate $$\int_{-\infty}^\infty\Gamma(1+ix)\Gamma(1-ix)dx$$Now here's the ...
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### Does using pre-computed squares speed up significantly the calculation of factorial $n!$?

There are many different methods that tried to improve the calculation of $n!$. Few of them managed to halve the number of mulitplications. One of those methods is the basis to completely remove the ...
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### Why does a sum of factorials behave differently from single factorials?

The Brocard Problem shows three factorials that can be expressed as $n!+1=m^2$ or equivalently $n!=k(k+2)$. So any time a single factorial can be put in the form of $k(k+2)$, it will belong to Brocard ...
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### How long before someone is ahead by 10 in a game of flipping a coin?

If me and a friend are playing a game and start with 10 marbles each. We flip a coin each round and each time it's a head he gives me a marble and each time it's tails I give him one. How many coin ...
### Binomial proof of $n! = n^r - n(n - 1)^r + \frac{n(n-1)}{2!}\left(n -2 \right)^r - \dots$ when $r = n$
Solve for $f(x)$ if $f(x)\cdot f(-x)=g(x)$. I have been having trouble figuring this out. I asked ChatGPT, and its answers don't work. I then looked online (I googled it), and was surprised to find ...