# 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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### Find the last two non-zero digits of $70!$

The question itself is quite straightforward, however, I am unable to get an exact answer to the problem. I have narrowed it down to four possibilities one from $\{18, 43, 68, 93\}$. The approach We ...
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### Formula for $\binom{a}k - \binom{a-x}k$ [closed]

Is there a formula for $\displaystyle\binom{a}k - \binom{a-x}k$? One that perhaps reduces it down to a single binomial coefficient multiplied by something?
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### Prove $\lim_{z \to \exp \left(\frac{2 \pi i p}{m !}\right)} \sum_{n = 0}^{\infty} {z}^{n !} = \infty$ (Whittaker-Watson)

The following is a problem posed by Lerch in 1885. Problem: For $m \in \mathbb{Z}$ and $p = 0 , 1 , 2 , \ldots , \left(m ! - 1\right)$. Show that \begin{align} \lim_{z \to \exp \left(\frac{2 \pi i p}{...
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### If $n,m$ are positive integers with gcd$(n,m)=1$, then $\frac{(m+n-1)!}{n!m!} \in \mathbb{Z}$ [duplicate]

If $n,m$ are positive integers with gcd$(n,m)=1$, then $$\frac{(m+n-1)!}{n!m!} \in \mathbb{Z}$$
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### Probability a factor is odd

Take the above question. In the solution it says there 18 factors of 2 in the number. But how do they work this out? I see no reasoning or intuition between the line of working 10 + 5 + 2 + 1 = 18. ...
1 vote
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### Bounding super exponential functions with factorial functions

I want to show that there exists polynomials $q(n)$, $p(n)$ with integer coefficeints such that $$(q(n)!)^{p(n)} > 2^{2^n}$$ for all $n \in \mathbb{N}^+$. Intuitively this inequality seems to ...
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### approximation for 'balls in bins' problem with upper restriction.

I am dealing with the famous problem of finding the how many integer non negative solutions there are for the equation: $x_1+\cdots +x_l=n$ with the restrictions $\forall i: 1\leq x_i \leq k.$ The ...
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### What is $r-n+1$ equal to? [closed]

So I am working on factorials and can easily show that $$r = \frac{r!}{(r-1)!}$$ By expanding the factorial $$r! = r(r-1)\cdots(r-n+1)$$ Then dividing both sides by $(r-1) \cdots (r-n+1)=(r-1)!$ ...
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### Closed form for $\Gamma(n+\frac{2}{3})$

I'm looking for a good closed form for $\Gamma(n+\frac{2}{3})$ involving at most $\Gamma(\frac{1}{3})$. I know that $\Gamma(n+\frac{1}{3})=\Gamma(\frac{1}{3})\frac{(3n-2)!!!}{3^n}$ and my question is, ...
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### Finite many primes for every positive integer $b$?

Consider the function $$f(a,b):=\sum_{j=0}^a (bj)!=1+b!+(2b)!+\cdots +(ab)!$$ Given a positive integer $b$ , are there always only a finite number of positive integers $a$ such that $f(a,b)$ is prime ...
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### with respect to which partial order is this f monotonic?

My first task is to define a fixpointoperator f. It should represent the factorial-function: $fac = fix f$ $f = \lambda\ g \rightarrow \lambda\ x \rightarrow if\ x == 1\ then\ 1\ else\ x * g(x -1)$ ...
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1 vote
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### proving an equation involving factorials is impossible

I have a calculation which ends up with the following form: $$a!(x!)^a=b!(y!)^b,$$ for $a,b,x,y$ non-negative integers. I would like to prove this is impossible in general unless $a=b$ and $x=y$ or ...
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### Sum of series in the form $\sum^{\infty}_{n=0}\frac{1}{(n+k)\cdot n!}$ for some k>0

I was recently given the series $$\sum^{\infty}_{n=0}\frac{1}{(n+3)\cdot n!}$$ and was told that it evaluated to a very nice number of $e-2$. However, I do not know how to arrive at that answer using ...
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1 vote
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Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e. $$X(\omega): = \sum_{A \in \mathfrak{A}} \mathbf{1}_{A}(\omega) <\infty$$ for all \omega\in \... • 23.7k 2 votes 1 answer 43 views ### Calculating a sum in the coefficient of some generating function I tried to calculate the coefficient of a generating function \begin{align*} \frac{1}{n} [u^{n-1}] e^{un} \frac{1}{(1-u)^2} \end{align*} and got to \begin{align*} \frac{1}{n} [u^{n-1}] e^{un} \frac{1}{... • 390 2 votes 2 answers 87 views ### Finding the maximumk$such that$(7!)!$is divisible by$(7!)^{k!}\cdot(6!)!$If$(7!)!$is divisible by$(7!)^{k!}\cdot(6!)!$, then what is the maximum value of$k$? At first glance I couldnt think of anything except Legendre's formula for calculating powers of a prime in a ... • 173 2 votes 0 answers 44 views ### Sum of factorials in 3D symmetrical random walk$\sum_{k=0}^n\binom{n}{k}^2\binom{2(n-k)}{n-k}\$

I'm trying to prove that in a tridimensional symmetrical random walk all states are transient. Since it's an irreducible Markov chain, I only have to prove it for a generic state, since they are all ...
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