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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2
votes
2answers
71 views

Why does this sum equal to (4^n -1)

How do I get to this solution? $\sum _{k=1}^n\left(\binom nk 3^{n-k}\right)=\left(4^n-1\right)$ I believe it's connected to this, which I know is true: $\sum \:_{k=1}^n\binom nk=2^n-1$
2
votes
2answers
77 views

Prove the inequality $n!\lt n^{n+\frac12} e^{-n+1}$ [closed]

Prove the following inequality: $$n!\lt n^{n+\frac12} e^{-n+1}.$$ Try to avoid induction if possible. Thanks!!
1
vote
0answers
130 views

The closed form of $\sum_{n=1}^{x}n!$

Let $$y=\sum_{n=1}^{x}n!$$ be the sum of consecutive factorials. What is closed form for $y$ in terms of $x$? Wolfram Alpha says that $$y=-(-1)^x\Gamma(x+2)(!(-x-2))-!(-1)-1$$ where $!x$ is ...
9
votes
1answer
303 views

Complex Factorial Equaling One

For what complex values of $z$ is $$z! =1? $$ Are they even all known? Are there finitely many or infinitely many? (Yes, the trivial $z$ are 0 and 1. )
2
votes
1answer
66 views

Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
1
vote
3answers
61 views

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite.

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite. I tried factoring it to show that there are two factors, thus composites but I can't figure out how to get rid of the ...
5
votes
1answer
80 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
9
votes
4answers
193 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
2
votes
0answers
69 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 ...
0
votes
3answers
84 views

Multiplication of the numbers $1$ to $n$

Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it? Something like the sumation of $1$ ...
0
votes
0answers
33 views

number of trailing zeroes of factorial raise to power by another factorial

Finding trailing zeroes in any factorial is easy. Every time you pass a multiple of 10 (or something 5 mod 10) you will accumulate another 0 For example 10! has two trailing zeros, one from ...
1
vote
1answer
54 views

Limit of factorial how to continue

$$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{(n+1)!-n!}}\right)=\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$ How to continue? the answer is $0$ ... thank you ...
4
votes
2answers
242 views

N! ends with exactly 30 zeros? [duplicate]

How many values of N exist, such that N! ends with exactly 30 zeros?
0
votes
1answer
50 views

Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, ...
6
votes
3answers
124 views

The relation between the number of $0$s which are at the end of $3^{n!}-1$ and that of $n!$

Let $a_n,b_n$ be the number of $0$s which are at the end of $3^{n!}-1,n!$ in the decimal system respectively. I found that $a_n=b_n+1$ holds for $n=4,5,\cdots, 10$. Then, my questions are... ...
1
vote
1answer
44 views

Permutations in circular arrangements

I have another permutation question that I'm having trouble with; this time with circular arrangements: To a meeting involving four companies, each company sends three representatives -- the ...
1
vote
1answer
17 views

Permutation/factorial question

I have this question: How many numbers greater than 40 000 can be formed using the digits 2, 3, 4, 5 and 6 if each digit is used only once in each number? The first digit needs to either be ...
0
votes
1answer
89 views

Given a prime p and an integer N, find the number of integers n such that 1≤n≤N and order(n!) is divisible by p

We are given a prime number $\leq 10^{18}$ and an integer N $(\leq N\leq 10^{18})$ how to find the number of integers lying in the range $1\leq n\leq N$ for which the order(n!) is a multiple of p? ...
0
votes
1answer
38 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq ...
4
votes
2answers
131 views

Limit of $a(k)$ in $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $

For n = 1, 2, 3 ... (natural number) $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $ $ a_1 = 1, \ a_2 = \frac{1}{2}, \ a_3 = \frac{7}{12} \cdots $ What is the limit of {$ a_k $} $ \lim_{k \to \infty} ...
1
vote
2answers
56 views

How can I show that $n^{n+2}<(2n)!$ for any integer $n$.

When I was try to show that the series $\sum_n \frac{n^n}{(2n)!}$ is convergent using comparison test, I stuck at the point $n^{n+2}<(2n)!$ I think it can be show using mathematical induction. If ...
1
vote
2answers
22 views

Counting Problem using Permutations

The Question was: In how many ways can the letters of the English alphabet be arranged so that there are exactly 10 letters between a and z? My approach was the following: In between a and z, there ...
0
votes
2answers
125 views

Integral of factorial function

What can we say about the integral $\displaystyle\int_{0}^{a} x! dx$? Or something like $\displaystyle\int_{0}^{3} x! dx$?
1
vote
1answer
69 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
1
vote
1answer
32 views

Using Stirling's approximiation to show that $(\log(\log n))!$ is $O(n^k)$

I am trying to show the following: Prove, using Stirling's approximiation, that $(\log(\log n))!$ is $O(n^k)$ for some positive constant $k$. Stirling's approximation is $$n!=\sqrt{2\pi ...
2
votes
1answer
53 views

A better approximation of $({n \over e})^n$ than by $ \Gamma(1+n-1/2)$ ? (Focus is “reverse” to the Stirling approximation)

In a formula in my self-study of a summation method based on the matrix of Eulerian numbers (which I thus call "Eulerian summation") I am considering terms like $$ \Big({n \over e}\Big)^n \cdot {1 ...
36
votes
3answers
3k views

Is $n! + 1$ often a prime?

Related to this question, I wonder how often $n!+1$ is a prime? There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not... or anything in that sense...
10
votes
4answers
883 views

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
-2
votes
1answer
30 views

Factorial and combinations question.

Any help with these would be greatly appreciated... 1) How many arrangements are there of the letters of the word SAUSAGES ? if the A’s must be together and the S’s apart? (answer apparently 240 ...
0
votes
1answer
28 views

Completely unique set in permutation

I have tried searching online for the answer and can't quite get one for my specific problem. My use of terminology is probably not helping (I don't study math). I think I know the answer but would ...
2
votes
1answer
148 views

Proof of an inequality involving factorials

How can the following inequality be proven? $$\left(n!\right)^{\frac{1}{n}}\left((n+1)!\right)^{-\frac{1}{n+1}}\gt\dfrac{n}{n+1}$$ I know this is a result obtained in 1964, but I don't know how to ...
0
votes
5answers
84 views

Factorial of zero is 1. Why? [duplicate]

Why is the factorial of zero, one. What is the mathematical proof behind it?
1
vote
4answers
102 views

Find $\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}$ [duplicate]

I am having trouble showing $$\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}=e.$$
3
votes
1answer
116 views

Euler proof of the formula involving factorial?

Let me be formal and write the formula Euler's Formula: Let $a$ and $n$ be nonnegative integers with $a\geq n.$ Then $$n!=a^n-\binom{n}{1}(a-1)^n+\binom{n}{2}(a-2)^n-\binom{n}{3}(a-3)^n+ > ...
4
votes
2answers
172 views

Euler's limit formula for the factorial function

In Euler's first (known) letter to Goldbach on October 13, 1729 he mentions an infinite product that interpolates the factorial function: ...
1
vote
5answers
174 views

Can the value of $(-9!)$ be found

I saw this question on an fb page and I couldn't solve it. Question: What is the value of $(-9!)$? a)$362800$ b)$-362800$ c) Can not be calculated The first options seems to be incorrect,which ...
1
vote
1answer
57 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
0
votes
0answers
38 views

Reasoning about factorials and powers of a finite set of primes

I am working on an answer to another question: How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$ I've reduced the question to showing that the following infinite set of ...
3
votes
4answers
149 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
1
vote
1answer
60 views

Factorial simplification

How can I work with this? $$\frac{(3n)!}{(3(n+1))!}$$ I really don't know how to open this fatorial and then, simplify it. Actually, I have to calculate the limit when $n\to\infty$. Thanks :)
2
votes
2answers
63 views

What is $\binom{a}b$ with $a<b$?

@Chris's_sis gave me following hint in a problem : $\frac{1}{ \displaystyle \binom{ p+k}{p}}- \frac{1}{ \displaystyle\binom{p+k+1}{p}} =\frac{ p}{p+1}\frac{ 1}{\displaystyle\binom{p-k-1}{p-1}}$ ...
1
vote
1answer
45 views

A quick question about factorials

So I'd like to write a function like this using factorials: f(x) = (x-1)(x-2) so that when I plug in x = 2 I get f(x) = 0. I tried this: f(x) = (x-1)!/(x-3)! which as I understand evaulates to ...
3
votes
0answers
58 views

Question about factorial function [duplicate]

Show that $$n!=1+\left(1−{1 \over 1!}\right)n+\left(1−{1 \over 1!}+ {1 \over 2!}\right)n(n−1)+\cdots$$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove ...
4
votes
3answers
113 views

Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$

Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I ...
2
votes
2answers
697 views

Limits with factorial

I'm having difficulties understanding all limits with factorial... Actually, what I don't understand is not the limit concept but how to simplify factorial... Example : $$\lim\limits_{n \to ...
0
votes
3answers
55 views

Proving by induction that $(n^2)!>(n!)^2$ for $n \geq 2$

I'm trying to prove that $(n^2)!>(n!)^2$ for $n \in [2,\infty) \cap\mathbb{Z^+}.$ Ok, here's what I've tried: $n \geq 2,$ $(n^2)!>(n!)^2$ ...
2
votes
0answers
45 views

Knuth shuffle : Is there a reciprocal to the factorial?

I have looked into the Knuth collection shuffle algorithm with pseudorandom number generators. They say that a PRNG with a seed state of $19937$ bits (like one of the Mersenne Twisters) can shuffle a ...
1
vote
2answers
41 views

prove that $N$ is divisible by $1,2,\ldots,k$ which $k+1$ is the lowest prime number after $N$

Suppose $n$ is a natural number ($n\ge 5$) and $k+1$ is the lowest prime number that is greater than $n$ prove that $A_i \mid n!$ which $A_i$ are these numbers: $1,2,\ldots,k$
2
votes
0answers
40 views

Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
0
votes
1answer
77 views

Find the largest integer $n$ such that $10^n$ divides $10^6!$

Let $N=10^6!$ Find the largest integer $n$ such that $10^n$ divides $N$. Furthermore, compute the first digit and the last non-zero digit of $N$. I have some ideas that you should be able to use ...