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
60 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
95 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
65 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
46 views

Inverse question of trailing zeros [duplicate]

$(5n)!$ has $2014$ trailing zeros. What is $n$?
-5
votes
2answers
42 views

Aptitude test question… [closed]

How does 4(4!)=24? (and not 96)
4
votes
0answers
62 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
354 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 ...
1
vote
4answers
50 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
2
votes
1answer
24 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
0
votes
4answers
33 views

Show $(2n+2)!\geq(n+2)(n+2)!$, $\forall n\in\mathbb{N}$.

It's the last step in a proof, and I just need to show that $$(2n+2)!\geq(n+2)(n+2)!$$ $\forall n\in\mathbb{N}$. I can't seem to do it though, any thoughts?
2
votes
2answers
41 views

Proof of factorial inequality concerning fractions

I'm having trouble with a proof, with the case $n>2$. THEOREM: For every natural number $n∈N$ where $n≠2$, $∑_{i=1}^ni≤n!$ Let us simplify the statement. ...
3
votes
4answers
100 views

Determine if this series $\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$ converges or diverges, and prove your answer?

Determine if this series $$\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$$ converges or diverges, and prove your answer? I've been able to prove similar problems, but I'm confused now that there's a ...
0
votes
1answer
33 views

Calculating p-adic valuation $v_p(n)$, using basic properties

Calculating p-adic valuation $v_p(n)$ I'm not confident with the properties of $v_p(n)$ Where $v_p(n) = $ the biggest integer $e$ such that $p^e$ divides $n$, if $n\not=0$, and $+\infty$ if $n=0$. ...
2
votes
3answers
41 views

Lower bound for the falling factorial $(2n)_{n}$

I'm looking for a lower bound for the falling factorial $$(2n)_{n}:= \frac{(2n)!}{n!}$$ I saw on Wikipedia that $n! > \sqrt{2{\pi}n}(\frac{n}{e})^n$ . So I assume that a possible lower bound ...
0
votes
2answers
64 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
vote
2answers
51 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
2answers
254 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.
0
votes
2answers
13 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
votes
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. ...
1
vote
2answers
24 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
votes
1answer
70 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
votes
2answers
496 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
votes
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
vote
1answer
31 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
vote
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 ...
0
votes
0answers
106 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 $x=3$. ...
0
votes
0answers
82 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... ...
1
vote
2answers
67 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
votes
0answers
49 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] ...
5
votes
6answers
167 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
votes
3answers
146 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
vote
1answer
41 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
votes
0answers
189 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
91 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
votes
1answer
53 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
votes
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 ...
0
votes
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
votes
6answers
93 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
votes
2answers
113 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
votes
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
votes
2answers
68 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?
1
vote
0answers
37 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
votes
1answer
52 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
votes
1answer
144 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
votes
1answer
70 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
vote
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
66 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
101 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
90 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. ...