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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3answers
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An identity involving the Pochhammer symbol

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a ...
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1answer
244 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
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1answer
77 views

Binomial coefficient properties

On "theoretical computer science cheat sheet" I found a special formula which is: $$ {n \choose k} = (-1)^k {k-n-1 \choose k}$$ But when I try to expand the value of ${k-n-1 \choose k}$ I have ...
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2answers
68 views

Finding $n$ in $n=ReverseFactorial(x)$ where $x$ is known.

My software application receives a series of very large integers (hundreds of decimal digits). So far, I have been using string/textual representation of decimal digits for very simple manipulation ...
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2answers
88 views

How many zeroes (at the end) will 10! have when written in base 3?

I can get the answer for the question by calculating 10! and then converting it to base 3 but is there a more logical point of view to this question which will mostly generalise the solution for this ...
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1answer
44 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
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4answers
373 views

How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$

Show that: $$\left(\dfrac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\dfrac{n}{n+1}\right)^n$$ where $n\in \Bbb N^{+}.$ If this inequality can be proved, then we have ...
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1answer
78 views

Find all possible values of r such that: nPr = r! [closed]

$$^{n}P_{r} = r!$$ Find all possible values of r.
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1answer
168 views

Sum of fraction of factorials

Can anybody explain this? $$\sum\limits_{k=1}^{\frac{m-1}2}\frac{(2k)!(2m-2k)!}{(2k-1)(2m-2k-1)k!^2(m-k)!^2}=\frac{(2m)!}{(2 m-1)m!^2}$$ I did actually simplify this to: ...
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3answers
100 views

Is There a Way to Specify Limits On a Factorial

If I want to be able to express a factorial -- let's say "20!" -- but with upper and lower limits such that the factorial is evaluated from Upper Limit, n1=20, through a Lower Limit, n2=10, for ...
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2answers
794 views

Factorials and Prime Factors

I need to write a program to input a number and output it's factorial in the form: $4!=(2^3)(3^1)$ $5!=(2^3)(3^1)(5^1)$ I'm now having trouble trying to figure out how could I take a number and get ...
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0answers
78 views

How to find other nontrivial solutions of $a!b!=c!$? [duplicate]

I know only one nontrivial solution of this equation: $6!\cdot 7!=10!$. There is also a series of trivial solutions: $n!(n!-1)!=(n!)!,\ \forall n\in\mathbb{N}$. So my question is how to find any other ...
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1answer
60 views

Simpler expression for $\sum_{k=1}^{n}{k!}$

Is there a way to express $$\sum_{k=1}^{n}{k!}$$ in a simpler way that doesn't use sums up to n ? I've searched for this around the web and found that the subfactorial function can help with this, ...
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2answers
135 views

Is there a factorial for factorials?

Is there a more succinct way to notate this? $$(n!)((n-1)!)((n-2)!)\cdots(2!)(1!)$$ for clarification, if I had asked a similar question, how to succinctly notate: $$(n)(n-1)(n-2)\cdots(2)(1)$$ I ...
3
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1answer
2k views

Infinite Sum of factorial denominator and exponential numerator

I've been trying to find the sum of the following infinite series: $$ \sum\limits_{n=1}^\infty \frac{x^n}{n!2^n} $$ I've rewritten it as $$\sum\limits_{n=1}^\infty \frac{y^n}{n!}, y=\frac{x}{2}$$ ...
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1answer
77 views

Limits and common sense

I'm stuck in understanding of limits. It all makes sense, but at a certain point my answers which seem logical to me are not true. Please can somebody explain why as a huge number gets divided by a ...
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7answers
202 views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without ...
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1answer
211 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 = ...
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1answer
48 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 ...
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1answer
56 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} ...
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1answer
124 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 ...
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3answers
174 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 ...
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1answer
66 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 ...
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2answers
1k 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 ...
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1answer
56 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: ...
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2answers
91 views

Showing $(n+1)^n<e^nn!$ by induction

Show $(n+1)^n<e^nn!$ I know why that would be the case using general knowledge and a bit of substitution but am clueless on how to prove it.
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6answers
204 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 ...
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2answers
145 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
118 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 ...
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2answers
856 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 have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
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2answers
117 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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3answers
117 views

Estimate the factorial $n!$ starting with the integral of $1/x$

This is a 3-part problem concerning an estimate for the factorial $n!$ a. By considering the graph of $y=\frac{1}{x}$ explain why $$\frac{1}{k+1} < \int\limits_{k}^{k+1} \frac{\mathrm ...
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2answers
72 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 ...
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4answers
80 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
79 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 ...
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1answer
92 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 ...
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1answer
60 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
188 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
54 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
248 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 ...
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2answers
141 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
60 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: ...
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1answer
36 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 ...
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3answers
653 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.
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0answers
81 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.
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2answers
60 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
105 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
58 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
68 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 ...
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1answer
708 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 ...