Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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51 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
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2answers
36 views

Solving Large Factorial Division without writing out factorials

I am calculating entropy for a physics problem and it requires solving this equation: $\ Entropy = \frac{949!}{899! 50!} $ However, I am not sure how to solve this mathematically without reverting ...
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2answers
290 views

Factorials in different base

Got an interesting problem from a friend. How many zeroes does $n!$ end in when written in base $n$? For every factor of $n$ in $n!$, I know that there will be $1$ $0$ added. However, I'm not really ...
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2answers
44 views

How do I simplify this equation

I'm trying to find a formula that will allow me to calculate the sum total of a progression (not sure if that's the word) in a spreadsheet. $$1 + 0.79 + 0.79\cdot 0.79 + 0.79\cdot 0.79\cdot 0.79 ...
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3answers
105 views

How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$

$$\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$$ Can someone show why this estimate holds true? I tried quite a bit but couldn't really find a way to approach this. WolframAlpha says it is true ...
3
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1answer
144 views

Uniquely identify any finite subset of an infinite set

Let $U$ be an unbounded subset of $\mathbb{N}$. Let $D = \mathcal{P}_{<\omega}(U)$ (the set of all finite subsets of $U$). Let $f$ be an injection such that: $f: D \rightarrow \mathbb{N} $ ...
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1answer
23 views

Factorial Series Written As Recursive Definition

I have a factorial series as shown below: \begin{equation} (2n+1)!~\text{for all $n \geq 0$} \end{equation} And I would like to know if the recursive definition that I wrote is accurate: ...
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0answers
110 views

Evaluate $\lim\limits_{n\to \infty} {\frac{(2n-1)!!}{(2n)!!}}$ [duplicate]

I have tried using Stolz–Cesàro's formula and subtract the next term in the series but that gave me $$\lim_{n\to \infty} {\frac{2n}{2n+1}}$$ which is obviously 1 and not right. I do realise that the ...
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4answers
115 views

Prove $n! \leq n^n$

Prove by least counter example for all positive integers n. $$ n! \leq n^n$$ I keep getting stuck after proving the least element of the set of counterexamples can not equal 1. Any suggestions would ...
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0answers
45 views

proof of multi choose equivalence

I could really use some help proving this. Let n and k be positive integers, and let $\left(\!\!{n\choose k}\!\!\right) ={n+k -1\choose k}$, prove: $$\left(\!\!{n\choose k}\!\!\right) = ...
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1answer
70 views

Hat check problem. Ten friends total, five with sombreros, five with fedoras.

A group of ten people give their hats to the coatroom attendant. Five of the ten are wearing sombreros, and five and wearing fedoras. How many ways can the clerk return the hats so that no one gets ...
5
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3answers
57 views

Is there an efficient way to calculate $a! \bmod b$ for arbitrarily large $a < b$?

I am trying to find a way to calculate $a! \bmod b$ where $0<a<b$ for arbitrarily large $a$. Is there an efficient way to do this without calculating $a!$ explicitly?
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1answer
46 views

Is there a formula to find the exact value of inverse factorials?

$x!=y, \space x ∈ ℝ$ Is there a formula to find the exact value of $x$ in this case, assuming that we know the value of $y$? I could do $L(x)/W(\frac{L(x)}{e}) + \frac{1}{2}$ where $W$ is the ...
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1answer
28 views

How many ways to arrange 10 values in a vector of length 50?

I have a set of values $1,2,3,4,5,6,7,8,9,10$ that I want to place in vectors of length 50 in all permutations. So far I'm using the permutation formula with $n=50$, $r=10$ as $$ \frac{50!}{(50-10)!} ...
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4answers
63 views

In layman's terms, what is a multifactorial? - $x\underbrace{!!\cdots!}_{n\text{ times} }$

My first impression was that given a multifactorial expression, one was to factorial the first term, then factorial that term, then factorial that term, etc. etc. So, 20!! = (20!)! I now understand ...
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3answers
84 views

How to find $\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}$

What method should I use for this limit? $$ \lim_{n\to \infty}{\frac{n^{n-1}}{n!}} $$ I tried ratio test but I ended with the ugly answer $$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}} $$ which ...
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0answers
28 views

Explaination of how Wallis's formula was used to arrive at the end result?

While looking for a proof on Stirling's formula I am having a little trouble connecting the dots between these last two steps here: [1]: ! The first formula is Wallis's formula.
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3answers
84 views

Limit as $n$ goes to infinity of $n^22^n/n!$ [closed]

I have read in a book that $( n^2 2^n)$ is superior to n! this means that the limit below will at least be a constant and $n! = O( n^2 2^n )$, but l could not manage to find it, any ideas ...
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2answers
50 views

Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
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3answers
57 views

How to simplify infinite multiple? [closed]

I have a series like $$1!2!3! ... n!$$ for some finite n. How do I write this in notation? Similar to summation I guess or if it can simply be reduced down to something in terms of n that would be ...
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2answers
70 views

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $ The way I am approaching this problem is just to find the ...
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2answers
45 views

Limit of n!^{1/n} two ways that give distinct results. Contradiction?

I tried two methods to compute the limit, $\lim \limits_{n \rightarrow \infty} \sqrt[n]{n!}$ and i got distinct results. Method 1: Observe that $(2n)! \ge \prod \limits_{k=n}^{2n} k \ge n^{n+1}$. ...
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28 views

Limit Ratio of Gamma Functions [duplicate]

How can one prove that for any $x>1$: $$\lim\limits_{n\to \infty }\left(\frac{n^x \Gamma \left(n+1\right)}{\Gamma \left(x+n+1\right)}\right)=1$$ It is easy to show this for $x$ a natural number ...
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1answer
92 views

Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
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0answers
28 views

Can the following sum involving double factorial be simplified?

I have the following sum: $$\sum_{k=2}^{n-1}\frac{(n-k)(2k-3)!!}{a^k}$$ where $p!!$ is the double factorial defined as $$p!!=\left\{\begin{array}{} p\cdot (p-2)\cdots 3\cdot 1, & \mbox{if $p$ is ...
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1answer
27 views

Is there a symbol for a factorial with a modular congruence condition?

I'm solving ODE's using power series and I'm often getting coefficientes that involves terms like \begin{equation} \prod_{n=1,\:\:n\not\equiv1\pmod{3}}^{3k}n=2\cdot 3\cdot 5\cdot 6\cdot 8\cdot 9\cdot ...
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1answer
96 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi ...
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3answers
68 views

Negative factorials and the creation of math functions

I know that factorials are, by definition, positive integers which means you can not have n! where n is negative. My question is can you create a factorial specifically for negative integers? I ...
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1answer
77 views

Evaluating a double sigma

Evaluate $$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{m!n!}{(m+n+2)!}$$ How do I start with the problem? Infinite sum of factorials?
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2answers
75 views

What does ${50}\choose{4}$ mean in statistics?

I have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...
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1answer
73 views

Prove n! < (n/2)^n by induction

Im supposed to prove that $n! \leq (\frac{n}{2})^n$ by induction and I got to know that its only valid for $ 6 \leq n$. I tried it solving this way: $6! \leq 3^6$ $720! \leq 729$ $(n+1)! \leq ...
2
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1answer
36 views

Asymptotic growth of products of powers of primes vs factorials

Suppose we are comparing products of powers of primes p vs. n!: p₁^(2n/p₁)∙p₂^(2n/p₂)∙p₃^(2n/p₃)∙p₄^(2n/p₄)∙... vs. n! If ...
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0answers
41 views

A factorial related inequality

Given $n$ is there an explicit or asymptotic formula for least $m$ such that $$m!\geq n?$$ Essentially is there a good inverse to factorial?
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23 views

Compute factorial given precision

I wonder if there is an algorithm that compute $n!$ (factorial of n) with a given precision: $d$ significant digits, and whose complexity is $O(d)$. The only way I know to compute an approximation of ...
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1answer
111 views

Factorials and their perfect squares

How many positive factorials are also perfect Squares. So for example $1!=1=1^2$. How many others exist other than 1? Is there any way to prove this?
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4answers
170 views

How many ways can I choose 5 items from 10?

I'm learning about factorials and it looks like choices (how many ways you can choose something) is related to factorial. Does anyone know how many ways can I choose 5 items from 10? (For example, 5 ...
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2answers
52 views

Proof (cases & induction): Find the set of positive integers such that $n! \geq n^3$

I need to find the set of positive integers such that $n! \geq n^3$, and then prove my answer is true using cases and induction on $n$. There is a lemma that I will need to prove and use for this ...
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1answer
44 views

Breaking apart factorials

Say you have a factorial like this in an equality: $\frac{(x-1)!}{((x-1)!-(y-1)!)!}$ Is there any way to split it apart? How can it be manipulated? The second factorial seems to complicate things.
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1answer
45 views

Using induction to prove the inequality $n!\ge 2^{n-1}$ [duplicate]

How would I go about proving that $n! \ge 2^{n-1}\ \forall n \ge 1$? The base case makes sense to me, but when I do the inductive step, I go here using the inductive step: $$ n+ 1 = k+1 $$ $$ (k+1)! ...
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1answer
29 views

Dividing factorial of a number using another factorial

I recently came across the following code that is used to calculate prime factorial ...
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26 views

Need help in following problems related to combinatorial analysis.

How many motorcycle number plates can be made if each plate contains 2 different letters followed by 3 different digits? How many four code words are possible using the letters in COMPUTE if (a) the ...
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2answers
90 views

Does series with factorials converge/diverge?

$$ \sum_{n=1}^\infty {{4^n n!n!}\over{(2n)!}}$$ I tried the ratio test but got that the limit is equal to 1, this tells me nothing of whether the series diverges or converges. if I didn't make any ...
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3answers
86 views

How many “words” of any length can be made from the letters in TREATS?

So, I feel as if I nearly understand this problem. It's not asking for actual words in the English language, just combinations of the letters. If not for the fact that this question says "of any ...
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1answer
46 views

Combinatorics Question. Id like a thorough explanation. Especially the part about multiplying or dividing each side and when its OK to do so.

The question is : ${n \choose 3} +{n \choose 2}+{n \choose 1}$. and it says to write the following expression without using the factorial symbol. Thanks in advance. The answer I got was $n^4 - \frac ...
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0answers
36 views

Divisibility of factorials

There are two numbers, $n$ and $p$, with prime $p$ and $n < p$. One is to calculate $n! \bmod p$. Is there any chance of doing this without explicitly determining $n!$ ? I already know that with ...
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1answer
58 views

Show $\sum^\infty (2n-3)!!/(2n)!!$ converges

I would like to compute the sum $\displaystyle\sum_{n=0}^\infty\dfrac{(2n-3)!!}{(2n)!!},$ where the double exclamation point refers to double factorial. Using double factorial identities we get the ...
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1answer
27 views

How to expand a factorial expression with $N$ and $m$

In a statistical physics book, I don't understand how they moved from this expression: $\Big(\frac{N}{2}-m\Big)! \Big(\frac{N}{2}+m\Big)! = \Big(\frac{N}{2}!\Big)^2$ with $N=2k, k,m\in ...
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2answers
47 views

Elementary proof of lower bound for factorial [closed]

Let $\xi$ be a fixed but arbitrary real number. Prove that $n! > \xi^n$ for sufficiently large $n$ in as elementary, short and elegant a way as possible. In particular, you are not allowed to use ...
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1answer
105 views

What is the use of Gamma function in real world?

Is there any real world use of this function and can it be helpful now or in feature? or is it just made for fun in mathematics? Im asking because i know that factorial it self is used for ...
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1answer
87 views

How to show that (2n-1)(2n-3)!! = (2n-1)!!

I need to show that $(2n-1)(2n-3)!! = (2n-1)!!$ in order to validate an expression for a proof. Wolfram Alpha tells me that this is true for $Re(n)=-1/2$ I don't know enough to know how to proceed in ...