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
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
42 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
89 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
219 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
47 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
33 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
69 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 ...
0
votes
1answer
28 views

Calculating factorial of a number in a specific base [closed]

How can I calculate factorial of a number of a certain base? for example factorial of 22 in base 3?
1
vote
3answers
44 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?
9
votes
1answer
150 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
50 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
705 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
83 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
77 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
558 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)?
5
votes
4answers
186 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
87 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
68 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
626 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
340 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?
1
vote
1answer
45 views

Is this equation for $2^k!$ correct?

I couldn't find any equation for $2^k!$ so I came up with an equation that appears to work for the factorial of a power of $2$. However, I'm having problems proving it. My equation: $$ \def\x{\times} ...
1
vote
3answers
86 views

How to prove that $\lim_{n \to\infty} \to \frac{(2n-1)!!}{(2n)!!}=0$

So guys, how can I evaluate and prove that $\lim_{n \to\infty} \to \frac{(2n-1)!!}{(2n)!!}=0$. Any ideas are welcomed.
5
votes
1answer
93 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are ...
0
votes
0answers
37 views

For which value of $n$ is the value $S_n = 1!+ 2! +\cdots+ n!$ a square of an integer? [duplicate]

For which values of $n$ is the value $S_n = 1!+ 2! + \cdots +n!$ a square of an integer? How do you compute this or if anyone wants to give me a hint?
0
votes
1answer
46 views

Smallest value of n such that the product $n!$ ends in at least 10 zeros.

What is the smallest value of $n$ such that the product $n!$ ends in at least 10 zeros? I tried to do this by multiplying each number but it didn't work. Please help.
4
votes
6answers
75 views

Product of r consecutive integers is divisible by r!

Well in a book i am reading it is given that you can also prove this by showing that Every prime factor is contained in $(n+r)!$ as often at least as it is contained in $n!r!$. How does this prove ...
1
vote
0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
6
votes
4answers
2k views

Factorial of 1,e+80

Recently I started being very fascinated in logistics, and out of the blue came the question into my head, what is the factorial of the amount of atoms in the observeable universe, which is said to be ...
3
votes
3answers
83 views

The sum $1!+2!+3!+…+2007!$ is not a perfect square

Today my teacher told me to prove this:- Prove that $1!+2!+3!+...+2007!=\sum_{n=1}^{2007}(n!) $ is neither a perfect square nor a perfect cube. Not getting any idea. Please help. Thanks in advance.
0
votes
4answers
112 views

Mathematical Proof of zero factorial is 1 . [duplicate]

I wanted to know is there any mathematical proof underlying 0! = 1 rather than qualitative analysis
1
vote
0answers
44 views

The Number of The 0's in a Factorial

I need to find that the number of the 0's at the end of the number is odd or even in a factorial. For example: $0! = 1$ (Even) $5! = 120 $ (Odd) $18! = 6402373705728000 $ (Odd) Dou you have any ...
11
votes
3answers
142 views

$1!+2!+\ldots+n!$ cannot be the square of a positive integer

I have to prove that $1!+2!+\ldots+n!$ cannot be the square of a positive integer, $\forall n\geq4$. I've tried to do this with induction, but I don't seem to reach any satisfactory conclusion. Any ...
0
votes
4answers
49 views

$(n+1)!>n^2, \forall n\ge 4$

The base case is clear, since if $n\gt 4, 5!=120\gt 25=5^2$ So assume $n=k$ which shows that $k!\gt k^2$. Then if $n=k+1$, $$(k+1)!=(k+1)k!$$ $$\gt (k+1)k^2 $$ Induction argument $$=k^3+k^2$$ $$\gt ...
-1
votes
3answers
115 views

How to interpret $(2n)!$

It's all in question: how to interpret the factorial from $2n$? Is $(2n)!$ equal to $n!\times n!$ ? The problem is in Combinations if the combinations is $\binom{2n}3$. P.S. The main problem is ...
2
votes
1answer
33 views

Definition of factorial function

Is there a definition of the factorial function (for natural numbers) in the language of ordered rings?
1
vote
0answers
64 views

Digits of $n$ factorial

With the notable exception of $0$, for large enough $n$, the digits in base $10$ for $n!$ seem pretty much uniformly distributed (I have also checked for other few bases $> 2$). Have anyone ...
2
votes
5answers
105 views

Hint in Proving that $n^2\le n!$

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
0
votes
1answer
26 views

AM I doing this right? - How many binary words of length 8 are there that contain at least six 1's?

How many binary words of length 8 are there that contain at least six 1's? This is what I have: 8!/6!2! = 28 words Is this the correct answer?
2
votes
3answers
186 views

Simplify the expression $(2n)!/(2n+2)!$

I'm a little confused as to how $(2n)!/(2n+2)!$ looks when written out. Basically I'm trying to visualise it so that I know how to cancel this and like terms in future.
0
votes
0answers
57 views

lower bound for factorials product

For $n$ positive integer, let $F(n) = 1! × 2! × 3! × 4! × \cdots × n!$, product of factorial(i) for $i$ in $[1\ldots n]$. Let $G(n) = \{i \in [1\ldots n],\text{ such that }n\mid F(i)\}$. It is ...
1
vote
1answer
158 views

Factorials in Sigma Notation

Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$. But how would I convert a sigma notation problem with ...
5
votes
2answers
59 views

The number of primes in the factorization of $N!$

Is there an approximation to the number of primes in the factorization of $N!$? For example: For $N=10$, this number is $15$. For $N=100$, this number is $239$. For $N=1000$, this number is $2877$. ...
1
vote
1answer
53 views

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$.

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$. I'm trying to prove the statement by building on my observation that $(1-\frac{1}{n})^n$ ...
0
votes
2answers
25 views

Factorial simplication question

How does the following: $$(k+1)! - 1 + (k+1).(k+1)!$$ simplify to: $$ (1+k+1).(k+1)! - 1 $$ and then $$(k+2)! - 1$$ I just can't seem to see how that works, I've tried writing out the factorials ...
0
votes
2answers
29 views

How can we prove this = 1 for all n

$\displaystyle n!-\sum_{k=1}^{n-1}k\cdot k!$ By computing this by hand for several small values of $n$ I can see that it is always equal to 1. But I can't see how to prove that.
1
vote
3answers
69 views

Factoring added factorials

How do I facilitate prime factorization without brute-forcing the 600+ digit number? For example, how would I factor (82! + 83! + 84!) ?
1
vote
1answer
68 views

Upper bound for $n!$

Let $a\in\mathbb{N}$. is there an upper bound be for the smallest n so that $n!>a$? It doesn't have to be a good upper bound, just something that works. Thanks.