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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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
25 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
87 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
54 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
71 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
70 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
55 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 ...
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1answer
32 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
40 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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0answers
20 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 ...
4
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1answer
103 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
84 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
48 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
40 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
41 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
28 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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2answers
21 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
62 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
68 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
41 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
22 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
51 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
21 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
45 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
79 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
50 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 ...
4
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2answers
73 views

Evaluation of $\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}$

I have problems evaluating the following limit: $$\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}$$ What causes problems in particular is that I am unsure how to ...
37
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3answers
494 views

Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
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1answer
44 views

Presenting Factorials as a sequence of multiplied numbers

When doing maths that involve factorials I am often not sure as to how I should present the. i.e. $$ n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1. $$ But what if $n$ was ...
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2answers
32 views

Approximate this ratio of factorials

I need to approximate the ratio $$ \frac{(2M)!}{(2M-N)!}\bigg/\frac{M!}{(M-N)!} $$ where $M$ and $N$ are huge, and $M\gg N$. I tried taking the logarithm, applying Stirling's formula, and taking the ...
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2answers
91 views

Find the limit $\lim_{n\to\infty} \frac{(3n)! \, e^n}{(2n)!\,n^n\,8^n}$

Find $$\lim_{n\to\infty} \frac{(3n)! \, e^n}{(2n)!\,n^n\,8^n}$$ I tried by simplifying $n!$ divided by $n! = 1$? What should I do next? I get then $3!e^n / 2!n^n8^n$
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4answers
541 views

Summing reciprocal logs of different bases

I recently took a math test that had the following problem: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ The sum is equal to 1. I ...
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1answer
29 views

How to find the value of convergent series

$$\sum_{n>=j}^{\infty }\frac{(1-\beta)^{n-j}*\lambda^n}{(n-j)!}$$ $j$ is a constant positive integer and $0<\beta<1$ also a constant. This series converges, but I want to know the ...
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2answers
45 views

$\sum_{k=1}^n(k!)(k^2+k+1)$ for $n=1,2,3…$ and obtain an expression in terms of $n$

Find a closed expression in terms of $n$. $$\sum_{k=1}^n(k!)(k^2+k+1); n=1,2,3...$$ Any idea about how to do this.. I'm a new to this so a little explanation would be helpful. Thanks in advance!
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1answer
52 views

approximate $\sum_{k=0}^n {n \choose k} k! n^{-k} \sim {\sqrt{\pi n}\over \sqrt{2}}$ [closed]

Show that as $ n \to \infty $ $$\sum_{k=0}^n {n \choose k} k! n^{-k} \sim {\sqrt{\pi n}\over \sqrt{2}}.$$
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2answers
101 views

Statistical probabilities to win the lottery

I'm trying to figure out what is the statistical probabilities to win the lottery. Let's assume that we pick 6 numbers out of 40 so : 1/40 × 1/39 × 1/38 × 1/37 × 1/36 × 1/35 = 1/2763633600 Up until ...
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1answer
22 views

Compute: binomial coefficients

Compute from Left-Side: $$ {2p \choose p} -{2p \choose p-1} = {(\frac {1}{p+1})} {2p \choose p}$$ This is the answer $$ ={2p \choose p} -{2p \choose p - 1}$$ $$=\left(\frac{(2p!)}{(p!) ...
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0answers
30 views

proof that $n! \leq (\frac{n+1}{2})^n$ [duplicate]

How do I prove the following statement using induction? $(n \in \mathbb{N})$ $$n! \leq (\frac{n+1}{2})^n$$ So for $n = 1$ it is true, since $1 \leq (\frac{2}{2})^1$. So now: Assumption: $n! \leq ...
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2answers
36 views

Proving some inequalities with factorials

I'm having some difficulty proving a few inequalities with factorials, and I was hoping someone could point me in the right direction. The first one is for $k\in\mathbb{N}$ and $0\leq t\leq 1$, that ...
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2answers
62 views

Induction proof of exponential and factorial inequality

I'm trying to find a proof for the following statement, using mathematical induction: $$ (\forall n\in \mathbb N-\{0\}) n^n \ge n! $$ But I always get to a dead-end. I've done the basis step, for ...
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1answer
55 views

Selecting a bag of six donuts from eight varieties

I am not really understanding how to solve this set of questions which are involving premutations and combinations, can someone explain question (a) or more? 4.4 How many ways are there to choose a ...
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3answers
33 views

Computing binomial coefficients

Compute from Left-Side: $$ {n \choose p} {n-p \choose k-p} = {n \choose k-p \quad p \quad n-k } = {n \choose k} {k \choose p}$$ So i started computing until: $$ ={n \choose p} {n-p \choose k-p}$$ ...
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1answer
34 views

Factorial minimisation

What methods might be appropriate to find the lowest integer bound for $t$ in the following inequality for a given $p$? $(2t+2)! > 2^{p+1}$ The best I can do is by using the inverse gamma ...
0
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1answer
58 views

Highest prime factor of factorial.

For a program I wrote, I used the property that the power of the highest prime factor of a factorial is always 1. I couldn't find anything about this, but it felt right. I can't prove it. Is my ...
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2answers
31 views

Are these expressions (Gamma function and binomial) identical for $n\in \mathbb Z$

For $n\in \mathbb Z$ and $n\ge 0$, prove that: $$\frac{2\sqrt{\pi}\,\Gamma(\frac{1}{2}+n)}{\Gamma(n+1)}=\frac{\pi}{2^{2n-1}}\binom{2n}{n}$$ I started to prove. We now that ...
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3answers
38 views

Proof by Induction (Inequality)

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
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3answers
26 views

Rearranging for $n$ with a factorial

For my maths course I need to prove that $n!/2^n$ tends to infinity as $n$ tends to infinity. For this I have to rearrange $n!/2^n > ∂$ so that it says $n > ...$. How to do this? Thanks in ...
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1answer
23 views

How to factor constants from a ratio of factorials.

Consider: $$ n_k = [\frac{(k-1)(1-\rho)}{1 + (1-\rho)k}][\frac{(k-2)(1-\rho)}{1 + (1-\rho)(k-1)}]... $$ Where $k \geq 1$ and $0 < \rho < 1$. My interpretation is: $$ ...
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1answer
64 views

Does $625!$ have $156$ zeros at the end?

Someone wrote that $625!$'s last $156$ digits are zeros because $125+25+5+1=156$. If it's true that $625!$ has $156$ zeros at the end, how does "$125+25+5+1=156$" prove it?
2
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0answers
42 views

Expression for gcd( ( 0! x n! ) , ( 1! x n-1!) , ( 2! x n-2! ) ,…(k!,n-k!) ) [closed]

Can we reduce computation of $\gcd (0!n!, 1!(n-1)!, 2!(n-2)!,\dots, k!(n-k)!)$? I tried and could not get a general expression for it. Please provide an explanation.