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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1answer
154 views

Factorial of $(7/2)!$ [duplicate]

It's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator. I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = ...
1
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1answer
25 views

Number of orders and combinations

I have just done these two questions and I have answers for them but I am not sure if they are correct. A jazz band is to give one concert in each of nine selected cities. Calculate the total ...
1
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1answer
39 views

Determinant of parametric function and $0!1!2!…n!$

As answer to this question, I trued to calculate the wronskian of: $$\left| \begin{array}{ccc} e^x & e^{2x} & ... & e^{nx}\\ e^x & 2e^{2x} & ...& ne^{nx} \\ e^x & 4e^{2x} ...
8
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1answer
120 views

What are the conditions for $n^2 \nmid(n-1)!$

Q: What are the conditions for $n^2 \nmid (n-1)!$, given that $2\le n \le 100$ and $n\in \mathbb{N}$? According to me the two conditions must be: 1. $n$ is a prime number (since the factorization ...
0
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3answers
94 views

Digit sum/product/properties of n!

How would one go about finding the digit sum/product/other properties of n!? If not for n!, at least for n too large for a calculator or computer to compute. (n>1000,let's say). EDIT: People who ...
-1
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1answer
64 views

Where do the two $a!$s come from?

I have \begin{align*} (2a)! &\equiv a! (-a) \dotsm (-3)(-2)(-1) \pmod p\\ & \equiv (-1)^a a!a!\pmod p\\ &\equiv (-1)^a a!^2\pmod p. \end{align*} The $(-1)$ is just to get the ...
3
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2answers
576 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
1
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1answer
45 views

Comparing factorials (From greatest to least)

Let's say we're asked to arrange these factorials in descending order: $1000!, \,\,700!\cdot300!,\,\,500!\cdot500!,\,\,600!\cdot300!\cdot100!$ For the first three, we could do: ...
3
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6answers
180 views

Proof of $0! = 1$ [duplicate]

I have been recently studying binomial theorem and in that we very frequently encounter factorials. But one queer thing which I found is $0!$. Even more queer is its value which is $0! = 1$. I ...
6
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2answers
134 views

Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
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3answers
100 views

How can I express a factorial as 10^y [closed]

I have $x = 120!$ How can I express this as $x=10^y$? Motivation for this question: I had a comment by a friend saying "120! = 10^200", and I wanted to make sure. It turns out I can still say that ...
6
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2answers
426 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I tried to do it using binomial theorem but that doesn't help. How will we do this? Please help.
2
votes
2answers
63 views

Why can't the factorion of n digit number exceed n*9!

"A factorion is an integer which is equal to the sum of factorials of its digits." I read from mathworld that the factorion of a $n$-digit number cannot exceed $n\cdot 9!$. Why is this? I mean what ...
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2answers
68 views

Whether m divides n! or not?

I have a big number ($n!$). And I want to know whether $n!$ dividable by $m$ or not. Well calculating $n!$ is not a good idea so I'm looking for another way. Example: $9$ divides $6!$ $27$ does ...
1
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4answers
73 views

How do these equate?

I need to evaluate the following $$\frac{(n+1)!}{(n+1)^{(n+1)}} * \frac{n^n}{n!}$$ It should come to $$(\frac{n}{n+1})^n$$ Currently, I only know that the $(n+1)!$ cancels with the $n!$ to make ...
2
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4answers
73 views

Binomial expansions and factorials

How to calculate $$\frac{n!}{n_1! n_2! n_3!}$$ where $n= n_1+n_2+n_3$ for higher numbers $n_1,n_2,n_3 \ge 100$? This problem raised while calculating the possible number of permutations to a given ...
4
votes
1answer
84 views

Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
1
vote
1answer
58 views

How does $(k+1)!(k+2)-1 = (k+2)!-1$?

I'm trying to do a proof by induction question and I'm at the very last part. Apparently $(k+1)!(k+2)-1 = (k+2)!-1$. I have checked using an online calculator. I don't understand why though.
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4answers
149 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
2
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1answer
37 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
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3answers
149 views

Factorial limit from gamma function calculation

I want to show that $$\lim_{n\rightarrow\infty}\dfrac{\Gamma\left(n+\frac12\right)}{\sqrt{n}\Gamma(n)}=1$$ Using the formula for $\Gamma\left(n+\frac12\right)$ here, it reduces to ...
3
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2answers
77 views

Bernstein polynomial looks like this: $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.Find it's $r$'th derivative.

Bernstein polynomials are defined like this $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.I need to prove that $r$'th derivative of it is equal to: ...
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2answers
56 views

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$ I tried to to solve it from the right side: ...
0
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1answer
30 views

Language to describe a number smaller than, but related to Bell number

I understand that the Bell number $B_n$ is the number of partitions of a set of size $n$. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient ...
10
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3answers
498 views

Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?

Which is greater among $300 !$ and $\sqrt {300^{300}}$ ? The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.
2
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0answers
70 views

Find all possible solutions!

Find solutions for $$^nP_r=s!$$ For $(n,r,s)\in \mathbb{N}$ I could find some trivial solutions $(6,3,5)~,~(1,1,1)$ etc.
2
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2answers
57 views

Logic of statement

I can see the mathematical implication but could not get the logic, why $5!$ is equal to $^6P_3$? Please help proving why both the expressions are equal without mathematical manipulation!In any case, ...
2
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1answer
101 views

Why doesn't $0! = 1$ in the context of this general term?

Is my instructor wrong to say that $\left\{0,\frac{1!}{4},\frac{2!}{9},\frac{3!}{16},\dots\right\} = \left\{\frac{(n-1)!}{n^2}\right\}$? My understanding is that at $n=1$, $\frac{(n-1)!}{n^2}$ should ...
2
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0answers
40 views

Combinatorics - find $n!$ using inclusion-exclusion [duplicate]

difficult question I need help with. We are asked to show that $n! = \sum_{k=0}^n (-1)^k\binom{n}{k}(n-k)^n$ There is also a hint "try to think of the number of permutations of n elements using ...
2
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3answers
57 views

Check the convergence of series

$$\sum _{n=1}^{\infty } \frac{\left(2 n^2-n+1\right)!}{3^{n^2+1}}$$ Trying to solve with sign d'Alembert, nothing comes out, and prevents the transformation of quadratic factorial reduction. Wolfram ...
0
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1answer
264 views

Last digits of factorial

Yes, this is an attempt to understand why my solution for Project Euler problem 160 isn't working. I hesitate to post my code lest I offer a solution to someone else. The problem is to find the last ...
6
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4answers
3k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
2
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1answer
85 views

Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

I have to check for which $x$ the series converges/diverges. $\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$ I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > ...
1
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1answer
95 views

In how many ways can five balls be chosen so that…

In how many ways can five balls be chosen so that (a) two are red and three are black? (b) three are red and two are black? out of $7$ black and $8$ red Should I use permutation? or ...
2
votes
3answers
85 views

Limit of sequence. with Factorial

Can't find the limit of this sequence : $$\frac{3^n(2n)!}{n!(2n)^n}$$ tried to solve this using the ratio test buy failed... need little help
5
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4answers
212 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
4
votes
2answers
83 views

Show that $p!$ and $(p - 1)! - 1$ are relatively prime

If $p$ is prime number, with $p>3$ Show that $p!$ and $(p - 1)! - 1$ are relatively prime. I tried $\text{gcd}\;(p!,(p-1)!-1)=d\Longrightarrow d\mid p!$ e $d\mid(p-1)!-1$ having ...
0
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3answers
131 views

how to solve factorial involving multiplication

I am trying to solve this question but not able to find any helpful material. It involves factorial with multiplications, $$\frac{8!}{5!}\cdot \frac{7!}{7!10!}$$ I tried crossing 8 and 5 and 7 with ...
0
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1answer
44 views

Review of an answer for finding a limit of a sequence

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {{n!} \over {(n + 1)(n + 2)...(2n)}} = {{n!} \over {{{(2n)!} \over {n!}}}} = \cr & {{n!n!} \over {2n!}} = {{n!} \over 2} = + \infty ...
0
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1answer
117 views

Limit of a fraction of double factorials

How can we show that $\begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*}$ where $\begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*}$ ...
2
votes
2answers
125 views

Convergence testing involving factorial and square root

I'm trying to find the convergence of this using the ratio test: $$\displaystyle\sum_{i=1}^{\infty}\dfrac{1}{\sqrt{t!}}.$$ But I'm getting no luck! Can anyone help? (sorry I've not quite ...
3
votes
2answers
83 views

Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$ [duplicate]

It basically says it all in the title. I tried solving the inequality using the bernoulli inequality somehow $$\dfrac{\displaystyle\sum_{k=0}^n\frac{1}{k!}}{(1+\frac{1}{n})^n}\geq 1,$$ but the ...
1
vote
1answer
31 views

Is there an actual expansion of the Gamma function's integral?

$$\int_0^{\infty} x^{t-1} e^{-x} \, \mathrm{d}x = (t-1)! = \Gamma (t)$$ Is the expression $(t-1)!$ the actual result of integrating the gamma integral? Meaning, if you were to compute the integral ...
3
votes
1answer
109 views

Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

Stirling's approximation of the factorial for even numbers is given by $$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$ Further, the Euler numbers grow quite rapidly for large ...
1
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1answer
62 views

Limit of a function not using Stirling's Approximation

I want to compute the following limit: $$\lim_{n\to\infty} \frac{\left(\frac{e}{F_{n+1}}\right)^{F_{n+1}} F_{n+1}!}{\left(\frac{e}{F_n}\right)^{F_n} F_n!},$$ where $F_n$ is the $n$th Fibonacci ...
1
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3answers
136 views

Solve Algebraical.ly $0.5=\dfrac{365!}{365^{n}(365-n)!} $

How does one go about solving this equation? Not sure how to approach this as no factorials will cancel out. Im sorry I meant $\dfrac{365!}{365^{n}(365-n)!}=0.5$.
0
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3answers
56 views

Is $\sum_{n=1}^{\infty}\frac{2^nn!}{(n+1)!}$ absolutely convergent?

I'm very uncomfortable with factorials just because I haven't done many of them. But my basic understanding is if I start with (for example) $(n+1)!$ then this is equivalent to $(n+1)*(n)$ and if it ...
2
votes
1answer
58 views

Proving a sequence with induction reasoning

I have an assignment which I am quite stuck on. The question is the following: function f: N to N is defined recursivly: ...
0
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1answer
34 views

Series — Coefficient Cn and Radius of Convergence

. I'm lost, and my textbook is failing me
3
votes
3answers
103 views

Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$

I need some help, showing that the left hand side is equivalent to the right hand side. I tried but I get stuck, I am not sure if I am on the right path. Here is my attempt: $C(2n,n+1) + C(2n,n)$ ...