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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2answers
66 views

Find the number of trailing zeroes. [duplicate]

Find the number of trailing zeroes. $k=1^1\times 2^2\times 3^3\times \cdots \times100^{100}$ It usually involves calculating number of $5$'s in $5^5\times 10^{10}\times 15^{15}\times \cdots\times ...
1
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2answers
71 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
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2answers
258 views

Why is it defined that $(-1)!!=1$?

Why is it defined that $(-1)!!$ equal to $1$, where $!!$ is the double factorial? I've only seen it defined that $(-1)!!=1$, but I don't see why it should be so.
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2answers
14 views

What's the difference between derangements and partial derangements?

What's the difference between derangements and partial derangements? I know that Derangements are essentially subfactorials; could anyone explain the difference? I came across this in some local ...
2
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1answer
42 views

$\frac{N!}{(N-n)!}$ when $n<<N$

I need to show that for $n<<N$ then $\frac{N!}{(N-n)!} \approx N^{n} $ I can see that $\frac{N!}{(N-n)!} = (N)(N-1)...(N-(n-1))$ and intuitively its clear but I am unable to show rigorously. ...
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2answers
25 views

Combination vs permutation

A teacher has $5$ books to distribute to some of $20$ children in her class. How many ways are there for her to distribute the books if the books are all the same and no child gets more than one? ...
4
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1answer
86 views

Finding missing digits in factorials

14!=871a82b1200 without working out 14!, find a and b I think it has something to do with maths rules regarding 9 or 3 (the digits adding up to either of those numbers) but not entirely sure!
9
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2answers
499 views

when is $n!+10$ a perfect square?

When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
2
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0answers
230 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...
1
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1answer
32 views

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$. Using the theorem ${n! \over k!(n-k)!} = {n \choose k}$ I simplified this down to $(a-c)! = ...
1
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1answer
34 views

Divisibility problem using Wilson's theorem: $4(p-3)! + 2$ is divisible by $p$

Prove that $4(p-3)! + 2$ is divisible by $p$, where $p$ is an odd prime. Use Wilson's theorem. I am having trouble trying to bring it in the form where Wilson's theorem can be applied. Any help ...
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0answers
111 views

Equation with Sum of Factorial and Subfactorial

I am interested in finding solutions to the following equation $$x! + !x = a^3$$ where $x$ and $a$ are natural numbers and $!x$ is the subfactorial of $x$. I've found the solutions $x=1$ and $x=3$. ...
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0answers
85 views

How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... ...
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2answers
73 views

Partitioning positive divisors of 100!

Is it possible to partition all positive divisors of 100! (including 1 and 100!) into 2 subsets so that each subset has the same number of integers and the product of all the divisors making up the ...
0
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0answers
53 views

Permutations Without Repetitions

Given the set [A,B,C,D] how many distinct ways can I order all four of the members of the set? I see distinct, as a unique set, therefore [A,B,C,D] and [D,C,B,A] ...
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6answers
168 views

The value of $ \int _{0}^{1}x^{99}(1-x)^{100}dx $ is

The value of $\int _{0}^{1}x^{99}(1-x)^{100}dx $ is Not able to do. I'm trying substituton. But clear failure. Please help.
0
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3answers
233 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 ...
1
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1answer
42 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
4
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0answers
191 views

Proving $11! + 1$ is prime

Prove that: $$11! + 1$$ is a prime number. Without computing the number (or factorial). Obviously, from Wilson's theorem, a number $n$ is prime if, $$(n-1)! + 1 \equiv 0 \pmod{n}$$ Since $n = ...
4
votes
2answers
92 views

Calculate $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!\cdot 2^n}$

Calculate the sum $$\displaystyle \sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!\cdot 2^n}$$ where $(2n-1)!!=1\cdot 3\cdots (2n-1)$, $(2n)!!=2\cdot 4 \cdots 2n$ Using Wolframalpha, the result is ...
0
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1answer
55 views

Is there any way to simplify this difference of factorials?

is there any way to simplify this expression or write it as a neat, concise formula? $$ \frac{(2m)!}{2m!} - \frac{(x+y)!}{x!y!} \cdot \frac{ [2m-(x+y)]!}{ (m-x)!(m-y)!} $$ Thank you!
3
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2answers
92 views

Stupid factorial question.

If i have $(3(n+1))!$ can I say: $(3(n+1))! = 3(n+1) \times (3n)!$ but if I expand by first multiplying the expression in the parenthesis; $(3(n+1))! = (3n+3) \times (3n+2) \times (3n+1) \times ...
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0answers
42 views

Is $\frac {(n^2)!}{(n-1)^2!} > n^2\#$ where $n^2\#$ is the primorial for $n^2$

Here's my thinking for why $\dfrac {(n^2)!}{(n-1)^2!} > n^2\#$: For $n=2$, $\dfrac{4!}{1!} = 24 > 4\# = 6$ Assume it is true for all $n$ so that $\dfrac{(n^2)!}{(n-1)^2!} > n^2\#$ ...
2
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6answers
99 views

Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 ...
3
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2answers
116 views

The ultimate formula to factor them all.

Context I am working on Integer factorization problem, I found a formula for factoring numbers, and I need your help to simplify it. First I will explain how I get there and then I present the ...
0
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2answers
47 views

A series specifying factorial [duplicate]

Is there any mathematical contraction of the following factorial series function? $$ F(n) = 1.1! + 2.2! + 3.3! + ...... + n.n! $$ I tried it by inspecting that $$ x.x! = x^2.(x-1)!$$ I need ...
0
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2answers
79 views

Prove that factorial grows faster than exponential function using limits [duplicate]

How can I prove that the factorial function ($n!$) grows faster than exponential functions (ex: $2^n$) using limits?
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0answers
38 views

Integral representation of simplified factorial division

I know that $\int_0^\infty{t^ne^{-t}}dt=n!$ from the gamma function, but I'm looking for a single integral which can represent $\frac{n!}{(n-m)!}=n(n-1)(n-2)\cdots(n-m+1)$ so that I don't need to ...
3
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1answer
55 views

Simple factorials

I've been doing some work with factorials and the normal way of calculating them is simply not working so well. When the numbers get really big, doing iterative multiplications is not viable and gets ...
6
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1answer
148 views

Prove $\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$

Let $T$ be the set of nonnegative integers, I need to prove that $$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$$ It's really easy to show that $$\limsup_{n\to\infty}|\{(p,q)\in T\times ...
11
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1answer
72 views

Does it follow that $(n!)^n$ divide $(n^2)!$

It is well known that $(n!)^2$ divides $(2n)!$. Does it follow that $(n!)^3$ divides $(3n)!$ and so on up to $(n!)^n$ dividing $(n^2)!$? If yes or no, could you provide the details behind the ...
1
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0answers
32 views

Rewriting Factorial Expressions [duplicate]

If I have an equation, say, $$ y = x! +1 $$ Is it possible to rewrite the equation in terms of $x$? For example, a simple algebraic equation might be $y = x^2$ --- and to rewrite it in terms of $x$, ...
5
votes
2answers
67 views

Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$

Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$ I tried using induction, but I failed in that approach, I rather found it was untrue, but on several case testing, I found ...
6
votes
3answers
105 views

Limit $\lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} $

First of all, sorry if something similar to this has been posted before (it's my first time in this web). I need to calculate the limit as $n\rightarrow \infty$ for this: $$\lim\limits_{n\to \infty} ...
1
vote
1answer
93 views

How to find the number of permutations with offset restriction

First question. Okay I have this problem that I've been trying to figure out for a while. I'm writing a computer program I need to quickly calculate the permutations of a set with 'n' elements with a ...
2
votes
2answers
144 views

Integer factorization: What is the meaning of $d^2 - kc = e^2$

I found an interesting behavior when placing the integer factorization problem in to geometry, I call it pyramid factoring. Lets assume we have $c$ boxes and we want to order them in to rectangle. ...
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2answers
64 views

Isn't $ n(n-1)(n-2)…(n-m+1) $ a factorial already?

Let $ m \ge 1 $ and $ n \ge 1 $ be integers Let $A$ be a set of size $m$ Let $B$ be a set of size $n$ How many one-to-one functions $f: A \rightarrow B$ are there? skipped stuff $$ ...
0
votes
1answer
56 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
0
votes
1answer
26 views

Series increasing or decreasing with factorials

I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are ...
0
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1answer
49 views

Computing $\mathrm{gcd} (100!, 3^{100})$

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem ...
0
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3answers
83 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
3
votes
1answer
49 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
1
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2answers
39 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
0
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1answer
74 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
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1answer
164 views

Finding the largest factorial with only three distinct decimal digits

I want to find the largest factorial whose decimal representation contains only three distinct digits. I am using the following Python code to compute the above, but no results up to 16000!: ...
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2answers
61 views

$x!=y^n$ for $x,y \neq 0,1$

A straightforward problem (find all integers such that $m!+3=n^2$) led me into thinking about the integers for which: $$x!=y^2$$ is true. I argued that other than the trivial case ($x!=1$) that this ...
1
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2answers
163 views

Inequality $(n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n$

Prove that $$ (n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n $$ holds for all $n\in\mathbb{Z^+}$. I tried induction but there's no obvious way to go from $n$ to $n+1$.
10
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3answers
335 views

Proving an identity involving factorials

I have stumbled upon the following statement and have verified it computationally for many $n$ (up to n=500, it took a long time for my computer to do out all of the math), yet I have no idea how to ...
0
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0answers
24 views

Function to define how combinations N items can be organized with a certain condition

This is not a factorial only problem If I have 5 items and I wanted to know how many possible ways they could be arranged, the answer is 5! or 120. However my situation is I need to know how many ...
0
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3answers
60 views

Simplifying factorials

I apologise for a repost, but my rep is not high enough to ask in a comment. But, in this question Simplify sum of factorials with mathematical induction I am confused how: $$(n+1)!-1+(n+1)(n+1)! = ...