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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4
votes
2answers
64 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
3
votes
1answer
37 views

Evaulate a determinant involving factorials.

In a problem set given by a teacher, there is the following problem. If $a_n = \frac{1}{n!}$, evaluate $$ D_n = \begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0 & 0 & ...
1
vote
3answers
98 views

Why is Wolfram Alpha miscomputing this problem? [closed]

I was incorporating Wolfram Alpha into an API I am build, and to test it entered a few equations. One of the equations I entered was as follows. !6/(!3*!3) This ...
6
votes
3answers
541 views

How do you find the factorial of a decimal or negative number and what does it show us?

I know that you can find the factorial of positive integers where n!= n(n-1)...2 x 1. However, what if you want to find the factorial of a negative integer or a decimal? I tried to do it on my ...
6
votes
1answer
66 views

Finding all the zeroes in $100!$

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers) I know that to find the $0$s at the end we can use the greatest integer method. I was just ...
10
votes
1answer
106 views

Last nonzero digit of $2010!$ [closed]

I have to calculate the last nonzero digit of $2010!$ Till now I couldn't find any pattern.
0
votes
3answers
44 views

Stirling Formula

Find the value of $\lambda$ for this question: $\dbinom{8n}{4n} \sim \lambda \dfrac{2^{8n}}{\sqrt{n}}$ as $n \to \infty$ I tried using Stirling. Any help appreciated.
1
vote
1answer
46 views

Squeeze Theorem for Factorials

I have been having trouble with questions with factorials in Squeeze Theorem. This is the questions that I am struggling with: $\lim_{x\to \infty} {x^x\over(2x)!}$ What I have done so far: Lower ...
0
votes
1answer
70 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
1
vote
1answer
45 views

Relationship between Factorial and Binomial coefficients

Over at this link, there is a claim that $(2n)! = n!n! {{2n} \choose {n}}$ - see Tom Boardman's answer, the second one down. I'm wondering why this is the case and if anyone can provide a proof. Is ...
0
votes
2answers
76 views

Need to show following equality

I want to show that the following equality holds for any integer i,m, and n.I could not figure out how to show it analytically. Could you please help me? $$ \sum _{j=0}^n ...
1
vote
3answers
69 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...
0
votes
2answers
72 views

Solving inequality equation involving sum of binomial coefficients

I have a function $f(k,\,i)$ involving binomial coefficients: $$f(k,\,i)\,=\left(\begin{matrix}k+i \\ k\end{matrix}\right)=\frac{(k+i)!}{k!\,i!}$$ And the following sum over this function (expansion ...
7
votes
4answers
158 views

What is the limit as k approaches infinity of $(k!)^{\frac{1}{k}}$ [duplicate]

What is the value of $$\lim_{k\to\infty}(k!)^{\frac{1}{k}}?$$ One of my students concluded the limit was infinity – which I tend to agree with, but was unable to show that was the limit. We ...
0
votes
1answer
37 views

Factorals with exponents. Is their a way?

I know of multiplication factorials with the 4! = 4*3*2*1 and I know of the addition with the nth triangle. I am busy deriving my own equation for something, and i am getting stuck on how to furthur ...
4
votes
3answers
90 views

Find $55! \bmod 61$

I am asked to find the smallest positive $x$ such that $x \equiv 55! \pmod{61}$. This invokes Wilson's theorem where $(p-1)! \equiv -1 \pmod p$. This means $60! \equiv -1 \pmod{61}$. But where to ...
3
votes
2answers
24 views

If $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$.

Using only precalculus knowledge, if $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$. (taken from Apostol's Calculus I, page 46) I don't ...
1
vote
2answers
58 views

In how many ways can you select one of the two but not both?

For this question: A committee of three boys and three girls is to be selected from a class of 14 boys and 17 girls. In how many ways can the committe be selected if: a.) Ana has to be on the ...
0
votes
0answers
19 views

What is the probability that a cluster of particles will contain some fraction of labeled particles, given total fraction of labeled particles?

Say you have a very large (but known) number of particles ($C$) and some known fraction of these particles are "labeled" ($F_L$). The particles spontaneously group into clusters of $n_c$ particles. ...
0
votes
2answers
107 views

Solve an equation involving factorial: $\frac{(n+1)!}{(n-2)!}=990$

For this equation: $$\frac{(n+1)!}{(n-2)!}=990$$ I need help with the working to the answer. Well I was stuck on the bit where I had ended up with: $$(n+1)n(n-1)=990$$ $$(n^2-1)n=990$$ ...
2
votes
2answers
46 views

Which rule is applied to define the operator precedence for factorial

Please apologize the question, I struggled with finding a good formulation in the first place: Looking at $\binom{2n}{k}$ it is very clear that for n,k integer and n>k we can solve it by calculating: ...
6
votes
1answer
92 views

Factorials: Simplifying $\frac{2017!+2014!}{2016!+2015!}$ to the nearest integer.

Compute $\dfrac{2017!+2014!}{2016!+2015!}$ to the nearest integer. My solution to the problem is 2016. Just wanted to check if it's correct. ...
3
votes
2answers
70 views

$\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$ [duplicate]

I know it's $\frac{\sqrt{\pi}}{2}$ but how can this be evaluated by hand? (Or can it not?) For quick reference: $$n!=\Gamma(n+1)$$ $$\Gamma(n)=(n-1)!$$ $$\Gamma(n)=\int_0^\infty x^{n-1} e^{-x}\,dx$$ ...
0
votes
0answers
10 views

Proving non base specific factorial trailing 0 counting function

I have come up with the expression below to calculate the trailing zeros of $x!$ when represented in base $B$ $$\left\lfloor\sum^{\left\lfloor\log_n x \right\rfloor}_{r=1} ...
1
vote
2answers
55 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
0
votes
1answer
38 views

Summation with factorial

I want to understand how this step is performed. Can you tell me that how this value of Po is obtained from the first equation.! ...
1
vote
6answers
107 views

Is $\frac{n}{3}! = (\frac{1}{3})^n n!$

Is $$\frac{n}{3}! = (\frac{1}{3})^n n!$$ I thought I could take all the (1/3) out of the factorial, but wolfram alpha says this is false.
1
vote
4answers
85 views

$\lim_{ n\to \infty} \frac{n!}{n^n}$ via L'Hospital's rule

I just need to find this limit and I don't know how to use L'Hopital's rule in this case: $$\lim_{ n\to \infty} \frac{n!}{n^n}.$$ I apologize for the lack of formatting, I've never used the site ...
0
votes
2answers
39 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
0
votes
1answer
40 views

Proving two equations containing ceiling and floor function to be equal

The problem is to show that if $n$ is a positive integer, then $$1 = C(n, 0) < C(n, 1) < ... < C(n, \left \lfloor{ n/2} \right \rfloor) = C(n, \left \lceil {n/2} \right \rceil) > ... > ...
2
votes
1answer
54 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
2
votes
8answers
139 views

How can $0!=1$ if the definition of factorial is $n!=n\times (n-1)!$ [duplicate]

Its a pretty basic question. If the definition of factorial is $n!= n\times(n-1)!$, then how can $0!=1$ since if we feed $0$ into the equation we get $0!=0\times (-1)!$? This comes after a ...
2
votes
2answers
81 views

Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$ n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this ...
1
vote
3answers
41 views

How to calculate $\lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$

I need to calculate limit number 1, and I don't understand how to get out the factors. $$ (1) \lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$$ $$ (2) \lim_{k \to \infty} \frac{(k)!}{(k+1)!}$$ When I ...
0
votes
1answer
35 views

function to approximate $x!$ without factorial

I am looking for a function $f(x)$ such that $f(x)\approx x!$, but (obviously) the function of x does not use factorial, eg a polynomial or exponential function. it does not have to be precise, just ...
0
votes
0answers
31 views

Coefficients for the falling factorial

Hello fellow mathematicians, I am trying to find a generating function, or at least find some useful property from the coefficients of the falling factorial. Let $(x)_n$ denote a falling factorial, ...
2
votes
1answer
41 views

integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$

Total no. of integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$ $\bf{My\; Try::}$ Let $w=\max\left\{x,y\right\}$. Then $w<z$. So we can write $w\leq (z-1)$ So ...
2
votes
1answer
38 views

Proof for 1/k! using n choose k as n approaches infinity and its relation to the gamma function

Prove that $\lim_{ n \to \infty }\binom{n}{k}(1/n)^k =\frac{1}{k!}$ How is this related to the gamma function?
4
votes
1answer
78 views

Combinatorial interpretation of double factorial.

Using some basic algebra (and proved afterwards using induction), I found that: $$ 1 \cdot 3 \cdot ... (2n-1) = \frac{(2n)!}{2^n \cdot n!}$$ After a bit of research, I found out that this is known ...
17
votes
4answers
470 views

Interpreting $n!$ as the volume of a $1 \times 2 \cdots \times n$ box

Q. Are there relationships or proofs that are illuminated by viewing $n!$ as the volume of a $1 \times 2 \cdots \times n$ box in $n$-dimensions? I cannot think of any, but perhaps they ...
2
votes
2answers
50 views

How can I express the ration of double factorials $\frac{(2n+1)!!}{(2n)!!}$ as a single factorial?

How can I change the double factorial of $$\frac{(2n+1)!!}{(2n)!!}$$ to single factorial?
0
votes
2answers
45 views

Show that $n! < (n/2)^n$ for all large enough $n$ in as elementary a way as possible

Show that $n! < (n/2)^n$ for large enough $n$ in as elementary a way as possible. Using Stirling's formula is not allowed. Of, course, what is true, is that $n! < (n/c)^n$ for any $c < e$ ...
3
votes
1answer
38 views

Expressing $\sum_{k=1}^{n}\frac{1}{(k+2)k!}$ in terms of $n$.

How would I express $$\sum_{k=1}^{n}\frac{1}{(k+2)k!}$$ in terms of $n$? An attempt of mine is $$\sum_{k=1}^{n}\frac{1}{(k+2)k!} = \sum_{k=1}^{n}\frac{1}{(k+1)! + k!},$$ which is not useful for ...
2
votes
1answer
97 views

Factorials…How do they do it?

So I've been recently arguing with my teacher about factorials. My teacher says that factorials can only be calculated for integers, because the definition of factorials is as follows: the product ...
2
votes
2answers
66 views

Summation over a product of binomial coefficients

Question: I can't figure out why the following equality is true $\sum_\limits{k=a-b-c}^{d} (-1)^k \binom{d}{k}\binom{k+b+c}{a} = (-1)^d \binom{b+c}{a-d} $ How can this be shown? (In the book it just ...
6
votes
2answers
110 views

Sum of factorial fractions

Find the sum $$\sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!}$$ I tried making something like a geometric series but couldn't. Then I couldn't think ...
2
votes
1answer
72 views

Floor function of a factorial

Compute $$\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.$$ How can I start with the problem? I thought of dividing by some number, but then I thought that some small numbers when added ...
-2
votes
1answer
51 views

Inverse question of trailing zeros [duplicate]

$(5n)!$ has $2014$ trailing zeros. What is $n$?
4
votes
0answers
71 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
5
votes
4answers
366 views

Easy Double Sums Question: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(m+n)!}$

How to calculate $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(m+n)!} $ ? I don't know how to approach it . Please help :) P.S.I am new to Double Sums and am not able to find any good sources ...