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
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2answers
72 views

Last digit of $235!^{69}$

Problem What is the last digit of $235!^{69}$? It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back. My initial thought goes to the last digit ...
3
votes
1answer
41 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
2
votes
2answers
48 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
93
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
5
votes
2answers
126 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
31
votes
4answers
32k views

How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a ...
13
votes
0answers
219 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
0
votes
2answers
38 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
0
votes
0answers
22 views

How to find the first and second number of factorial

How to find the first and second number of $40!$ Example $8!=40320$ the first is $4$ and second is $0$ I want to see the solution
0
votes
4answers
2k views

Gamma function proof of gamma $\;Γ(1/2) = \sqrt \pi\;$

So our teacher doesnt use the same demonstration as most other sites use for proving that gamma of a half is the square root of pi. I dont understand the demonstration from the first step because he ...
4
votes
5answers
153 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
0
votes
2answers
72 views

The method of solving for a factor of $90!$ [duplicate]

If $90! = (90)(89)(88)...(2)(1)$, then what is the exponent of the highest power of $2$ which will divide $90!$ ? How would I apply one of the easiest method from Here? I need help on applying ...
1
vote
4answers
63 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
4
votes
0answers
42 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
6
votes
4answers
832 views

Is $n \choose k$ defined when $k < 0$? What about $n < k$?

I know that ${n \choose 0} = 1$, and this makes sense to me based on my understanding of combinatorics. But what about ${n \choose -1}$? My instinct is that this is undefined, since it is equivalent ...
2
votes
0answers
29 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
4
votes
2answers
3k views

Proof the inequality $n! \geq 2^n$ by induction

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the ...
26
votes
15answers
6k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
25
votes
11answers
8k views

Division of Factorials

I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts? As a quick example $\frac{9!}{(2!3!4!)} ...
5
votes
3answers
517 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
1
vote
2answers
34 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
2
votes
4answers
78 views

Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
0
votes
2answers
28 views

Factorial with names

Ok so, I have had an argument with my teacher over 1 quiz question that was marked wrong in my data management class. Question. Determine the number of ways that 12 members of the boys' baseball team ...
0
votes
1answer
45 views

Why is 0 factorial 1? [duplicate]

n factorial is product of all numbers between n and 1. 0 factorial is (0 * 1 = 0). Why is 0 factorial 1? How can I proof this in mathematical way?
0
votes
0answers
38 views

Limit of a sequence $a_n = \sqrt[n]{n!}/n$ [duplicate]

Find the limit of the sequence $$a_n = \frac{\sqrt[n]{n!}}{n}$$ I can figure out the limit of the sequence by letting $n=1,2,3,\dots$ but what would be the more conceptual approach to finding the ...
0
votes
0answers
16 views

Solving $(\prod^t_{i=1} N_i m_i)!$

I would like to know how to solve or simplify the factorial $(\prod^t_{i=1} N_i m_i)!$. Here, $i, N_i, m_i$ are positive integers. My effort: $$(\prod^t_{i=1} N_i m_i)!$$ $$\implies (\prod^t_{i=1} ...
3
votes
1answer
90 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
2
votes
2answers
54 views

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$. I begin by letting $n=1$ then $\frac{1}{2}<\frac{1}{\sqrt{3}}$. Then assume $\frac{1\cdot ...
1
vote
5answers
454 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
27
votes
9answers
7k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...
33
votes
13answers
13k views

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is ...
0
votes
0answers
22 views

Find $\lim_{n \to \infty} \frac{2^n}{n!}$ - Stirling [duplicate]

Find $\lim_{n \to \infty} \frac{2^n}{n!}$. How is possible to solve this limite with Stirling formula? We can solve it with the ratio test, but I asked myself if it's possible with Stirling.
1
vote
4answers
133 views

Formula for factorial?

I need an equation that defines factorial without using factorial, that also works for $0$. I have seen factorial defined like this: $$n! = 1\cdot2\cdot3\cdot4\cdots n$$ But if we plug $0$ into that, ...
0
votes
1answer
23 views

Factorial grow faster than Exponential - permutation case

It is said that factorial grows faster than exponential, but in the case of permutation: ...
3
votes
5answers
605 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?
5
votes
1answer
68 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
50
votes
11answers
34k views
0
votes
0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
2
votes
2answers
491 views

Convergence testing involving factorial and square root

I'm trying to find the convergence of this using the ratio test: $$\displaystyle\sum_{i=1}^{\infty}\dfrac{1}{\sqrt{t!}}.$$ But I'm getting no luck! Can anyone help? (sorry I've not quite ...
0
votes
2answers
486 views

Powerball odds - factorial?

According to Powerball.com, the game is played like this ...we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls Their odds explain that the ...
2
votes
1answer
38 views

An inequality involving factorials and two variables

The problem is as follows: For $m\ge n>1$ prove that $$(m-2)!(n-1)+(n-2)!(m-1)+(m-2)(n-2)\ge (m-1)(n-1)$$ Since $(m-1)(n-1)-(m-2)(n-2)=m+n-3$ so we only need to show that ...
1
vote
1answer
78 views

Limit of factorials: $\lim_{n\to \infty}\frac {(2n-1)!}{2^{2n-1}n!(n-1)!}$

I'm failing go figure out how to calculate the limit where I have one factorial divided by two at about half its size. The specific limit I'm trying to find is this: $$\lim_{n\to \infty}\frac ...
8
votes
3answers
105 views

Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
0
votes
2answers
48 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. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
3
votes
2answers
126 views

Proving $r!$ divides the product of r succesive positive integers

I have to prove the following theorem: Prove that the product of $r$ consecutive positive integers in divisible by $r!$ I am having a hard time getting a generalization down for the full set of ...
1
vote
3answers
58 views

How to isolate variable algebraically in a combinatorics equation?

How would I isolate a variable algebraically in a combinatorics equation? For example, if I'm given: $$C(k, 2) = 45$$ How would I solve for $k$, without trying random values of $k$? I know that ...
6
votes
1answer
76 views

When $n!=m(m+1)(m+2)$: A Diophantine Equation

I believe that I saw this problem not long ago in a book: Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers. However, I am no longer able to find this question, and ...
4
votes
1answer
125 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...