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

learn more… | top users | synonyms

6
votes
3answers
131 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
81 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
21 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
99 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
39 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
135 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
59 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
25 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
148 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
74 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
42 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
66 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 ...
38
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. Does anybody know more about it?
10
votes
4answers
906 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
31 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
32 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
176 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
89 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
111 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
138 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
245 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
213 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
60 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
44 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
180 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
69 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
66 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
46 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
59 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
126 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
2k 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 ...
1
vote
3answers
69 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
55 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
44 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: $1,2,\ldots,k$
3
votes
0answers
45 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
79 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 ...
1
vote
3answers
45 views

Proof that $\lim\limits_{h \to \infty} \frac{h!}{h^k(h-k)!}=1 $ for any $ k $

I kind of barely understand this in some way, and I think I would understand it better by a formal proof. Where do I start?
5
votes
1answer
192 views

To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k , k\in \mathbb Z^+$ ?
10
votes
1answer
224 views

One-Line Proof for $n! \geq (\frac n e)^n$

I was told to find a one-line proof for $n! \geq (\frac n e)^n$. I'm advised that Stirling's formula is not helpful. I've spent a little bit of time on it, but the solution is not coming to me. I feel ...
6
votes
1answer
70 views

maximize a function which contains factorials

Suppose I have a function $$ f(k) = \binom{500}{k} \binom{500}{1100-3k}$$ where $k$ is an integer from $200$ to $366$. How can I find the maximum analytically?
11
votes
3answers
771 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
2
votes
2answers
90 views

How can I determine convergence of the series of $\frac {(2n)!}{2^{2n}(n!)^2}$

Does the series $$\sum_{n = 1} ^ {\infty} \frac {(2n)!}{2^{2n}(n!)^2}$$ converge? The ratio test doesn't work for the series.
0
votes
2answers
106 views

An identity for the product of even numbers (double factorial)

I'm unable to prove this identity: Prove that: $2\cdot 4 \cdot 6 \cdot 8 \cdots 2n = 2^n \cdot n!$ Wouldnt it be like this? $ 2(1 \cdot 2\cdot 3\cdot 4 \cdots n)= 2 \cdot n!$
6
votes
2answers
565 views

How can I unfactorilize number?

Consider the equation $x! = y$ Say we know $y$ and were trying to find $x$: What method could I use to get $x$ (e.g. a closed formula)?
6
votes
4answers
221 views

Direct proof that $n!$ divides $(n+1)(n+2)\cdots(2n)$

I've recently run across a false direct proof that $n!$ divides $(n+1)(n+2)\cdots (2n)$ here on math.stackexchange. The proof is here prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer (it ...
0
votes
2answers
105 views

How does $n!^2$ divide $(2n)!$? [duplicate]

How can I show that $(n!)^2$ divides $(2n)!$, where $n$ is a natural number? So far I've noticed that we can rewrite $\dfrac{(2n)!}{(n)!^2}$ as a combination and we know that combinations are always ...
0
votes
5answers
72 views

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$ I've already considered using l'Hoptials rules but I cannot take the derivative of a factorial (as it is a discrete function). Thanks
8
votes
3answers
1k views

What remainder does 34! leave when divided by 71 ??

What is the remainder of $ \frac{34!}{71} $? Is there an objective way of solving this? I came across a solution which straight away starts by stating that $69!$ mod $71$ equals $1$ and I lost it ...
2
votes
2answers
343 views

Is there a name for $\frac{n!}{m!}$?

Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?
3
votes
9answers
301 views

How can one prove $\lim \frac{1}{(n!)^{\frac 1 n}} = 0$?

I have tried bounding the terms by $\dfrac 1 {2^{\frac 1 n}}$, but this clearly cannot be made as small as possible.