Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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28 views

Maximum of a discrete function

I am trying to find the $k$ that maximizes of $f:\mathbb{Z}_{+}\to\mathbb{R}$ given by $f(k)= k^{-n} \frac{k!}{(k-m)!} \mathbf{1}_{k\geq m}$ as a function of $n$ and $m$, both positive integers with $...
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How to solve combination riddle question with a given scenario

A combination lock has 20 numbers. To open the lock, rotate clockwise to the first number, counter-clockwise past the first number to the second number and then clockwise to the third number. How many ...
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Factorial number theory question [closed]

What are the last three digits of the sum $1!+2!+3!+4!+.....2020! \;$? I got $313$.
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Is there any known value for $\sum_{n=1}^{\infty}\frac{1}{1+n!}$?

I just became curious about this sequence, so I tried several method to get exact value of the sum. I know it converges, but I have no idea to calculate the exact value. Is the value of $\sum_{n=1}^{\...
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30 views

Prove an inequality involving factorials and double factorials

Prove (or disprove) that for arbitrary non-negative integers $ m_e, m_f, M_e, M_f, $ such that $m_e + M_f \in 2 \mathbb{N}$, $M_e + m_f \in 2 \mathbb{N}$, the following inequality holds, $$ \frac{(\...
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2answers
70 views

Differentiation involving sigma notation

I am having trouble understanding the following relationship in one of my assigned problems: $$\dfrac{d}{dx}\sum_{n = 0}^\infty \dfrac{x^n}{(n + 1)!} = \sum_{n = 1}^\infty \dfrac{nx^{n-1}}{(n + 1)!}$$ ...
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1answer
82 views

To find all integer solutions using elementary number theory or linear algebra

Find all $(x,y,z)\in\mathbb Z^+$ such that $$\frac{x!+y!}{z!}=3^z$$ I was unable to find a nice way to approach the problem. I tried to use the fact that $\frac{x!+y!}{z!}\equiv 1\pmod 2$ but I got ...
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1answer
35 views

Write, with four fours and mathematical signs, an expression that is equal to a given integer.

Write, with four fours and mathematical signs, an expression that is equal to a given integer. The expression cannot include (in addition to the four fours) any number, or letter, or algebraic symbol ...
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1answer
50 views

Can one simplify this expression involving products of binomial coefficients?

I was wondering if there is a way to simplify the following expression. $N, M, L, K, n, Q$ are fixed natural numbers. Also n is supposed to be even. \begin{align*} \sum_{q=0}^Q & \sum_{l=0}^L (-1)^...
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3answers
96 views

What is the value of $\int_0^{\infty}\frac{1}{x!}\,dx$

I'm trying to figure out $\int_0^{\infty}\frac{1}{x!}\,dx$, but with no success, I tried approximating using Simpson's rule but i got $\frac{4e^2-e+2}{6e}$ which is about 1.76814... but wolframalpha ...
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1answer
27 views

perfect factors from the prime factorization of a large number

This is probably an easy question, but I don't know how to do it: In the prime factorization of $30!$, how many perfect factors occur? This is from a timed competition, any answers that take more than ...
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Depleted batteries

Accidentally, two depleted batteries got into a set of five batteries. To remove the two depleted batteries, the batteries are tested one by one in a random order. Let the random variable $X$ denote ...
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Definition of a Factorial

Now I have always been rather intrigued with factorial, at first, in high school, teachers told me that factorials are only defined for whole numbers. As I studied, I found factorials for positive ...
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Division between factorials

A jar contains five blue balls and five red balls. You roll a fair die once. Next you randomly draw (without replacement) as many balls from the jar as the number of points you have rolled with the ...
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2answers
20 views

Factorial to exponential conversion

I'm solving a statistical mechanics problem and in the solution, they have directly replaced $$\left (\dfrac{Nd}{2}\right)! =\left (\dfrac{Nd}{2e}\right)^{Nd/2}$$ Can someone tell me how to derive ...
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70 views

Prove that $\frac{2} {(0! + 1! + 2!)} + \frac{3}{( 1! + 2! + 3!)} + · · · + \frac{n} {(n − 2)! + (n − 1)! + n! )}= 1 − \frac{1}{n!}$

My attempts: By observation, we understand that P(1) is not defined. Hence we will prove this statement for $n\geq 2.$ P(2):L.H.S $= 2/0!+1!+2!=1/2$. R.H.S $= 1-1/2!=1/2$. Therefore, P(2) is true. Let ...
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65 views

Is there another way to write the product $ 2^2\times 3^2\times 4^2\times 5^2\times 6^2\times \cdots =(n^2)!\;$?

I have this product $$ 2^2\times 3^2\times 4^2\times 5^2\times 6^2\times \cdots =(n^2)!$$ Can we write this in another way? Thank you
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Find the number of trailing zeros of $ 100! - 101! + … - 109! + 110!$ [duplicate]

My try : I know number of trailing zeros of individual factorials, owing to their numbers of factors divisible by $5, 25$. ...
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4answers
94 views

How to solve $ \sum_{i=1}^{n} \left \lfloor{\log{i}}\right \rfloor $ for closed form

I'm trying to get a closed form of this equation: $$ \sum_{i=1}^{n} \left \lfloor{\log{i}}\right \rfloor $$ I know that $$ \sum_{i=1}^{n} {\log{i}} = \log{n!}$$ But I'm confused about how the floor ...
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3answers
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Is there a general way to find large enough $n \in \mathbb N$ such that for $k \in \mathbb N$: $n!>k^{n}$

I have found $n \in \mathbb N$ large enough such that $n!>2^{n}$ and $n!>5^{n}$ by simply "plugging in values". These facts have led me to consider a more general solution of $n!>k^{...
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86 views

Prove that $n$ and $n + k$ are both primes if and only if $(k!)^2[(n - 1)! + 1] + n(k! - 1)(k - 1)! \equiv 0 \mod n(n + k)$.

Prove that positive integers $n$ and $n + k$, where $n > k$ and $2 \mid k$, are both primes iff $$(k!)^2[(n - 1)! + 1] + n(k! - 1)(k - 1)! \equiv 0 \mod n(n + k)$$ According to Wilson's theorem, ...
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94 views

A sum of absolute values of binomial coefficients

Let $p>2$ be a real number and consider the sum $J=\sum_{d=0}^{\infty}|\binom{p-2}{d}|$. I want to know whether $J$ is a finite quantity or not? Indeed, if we consider $I=\sum_{d=0}^{\infty}\binom{...
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887 views

Is either $n! + 1$ or $n! - 1$ not prime for all $n$?

I was looking at an article about factorial primes, and I noticed that both $n!+1$ and $n!-1$ were not prime. (As in, there are no numbers $n$ such that both $n!+1$ and $n!-1$ are prime). I think that ...
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56 views

How to calculate this binomial sum?

In a previous post i was asking about a complicated sum which seems not possible to simplify: Is it possible to simplify this sum? I want now to calculate the simpler sum: $$ S := \sum_{k=1}^{n} \...
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80 views

Is it possible to simplify this sum?

I need help to simplify this sum: $$ \sum_{k=1}^{n} \binom{n}{k} ((k-1)!)^2 \left[ \binom{n}{k} - \binom{n-k}{k} \right]$$ I have tried to use Pascal's formula for the difference: $$\binom{n}{k} - \...
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50 views

When will i get a number that ends in $6 \cdot 10^n$?

I am attempting Brocards problem, and I have it pinned down to one simple question: When will $n(n-1)(n-2)(n-3)...6$ end in $6\cdot(10^n)$? So far I only have when $n$ is $6, 8, 14,$ and $19$. I ...
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Can anyone explain me how to find the value of $\left(\frac{1}{3}\right)!$?

I know how to calculate the value of $\left(\dfrac{1}{2}\right)!$ using the gamma function, but I don't know how to find the value of $\left(\dfrac{1}{3}\right)!$ or $\left(\dfrac{1}{6}\right)!$ using ...
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2answers
94 views

Mathematical induction $n! > n^{\frac n 2}$ [closed]

This problem from Lyashko I.I. et al. "AntiDemidovich: higher mathematic reference book", vol. $1$, ch. $1$ "Introduction to analysis" (ISBN $978-5-9710-7384-0$) Problem 31.a) ...
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1answer
65 views

Given that $2017$ is prime, how do I prove this statement?

I'm asked to prove the following statement: Let $N=(1008!)^2+1$. Prove that $N$ is divisible by $2017$. (Hint: $2017$ is prime.) I don't know how to go about proving this statement, since there seems ...
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40 views

Inductive proof that combinations are integral [duplicate]

Full question: Prove by induction on $n$ that: $$(n-r)!r!\mid n!$$ for $ 0\leq r \leq n$ for all $n \in \mathbb{N} $ I am having difficulty establishing a link to the assumption in the base case. ...
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1answer
87 views

Find $\sum_{r=1}^{20} (-1)^r\frac{r^2+r+1}{r!}$.

Calculate $$\sum_{r=1}^{20} (-1)^r\frac{r^2+r+1}{r!}\,.$$ I broke the sum into partial fractions and after writing 3-4 terms of the sequence I could see that it cancels but I wasn't able to arrive at ...
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102 views

How to solve $\sqrt{x!y!}=xy$ for $(x,y)\in\mathbb{Z}_{\geq0}\times\mathbb{Z}_{\geq0}$?

How to solve $\sqrt{x!y!}=xy$ for $(x,y)\in\mathbb{Z}_{\geq0}\times\mathbb{Z}_{\geq0}$? In this task they are asking to find ordered pair couple in $\mathbb{Z}$ that satisfies the above equation, So ...
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1answer
80 views

Will these observations be of any help in calculating $n!$ faster?

Well, hello. There's a recent observation I had when trying to figure out an easy and time-saving way to calculate $n !$: If $f(x) = 4x^2 + 2x $, $$ f(1)\cdot f(2)\cdot f(3)\cdot \dots \cdot f(n) = ...
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1answer
40 views

How many ways are there for 5 men and 11 women to stand in a line where there are at least 2 men in a row? [closed]

I have a question from a textbook that I could not really understand what's going on. The question goes like this: How many ways are there for 5 men and 11 women to stand in a line where there are at ...
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5answers
82 views

Evaluating $\lim\left(\frac{(2n)!}{n!n^n}\right)^{1/n}$

How to find: $$\lim_{n\to \infty}\left(\frac{(2n)!}{n!n^n}\right)^{\frac1n}$$ I tried as $\displaystyle \lim\left(\frac{(2n)!}{n!n^n}\right)^{1/n}=\lim_{n \to \infty}\left(\frac{2^n(1.3.5...2n-1)}{n^n}...
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How can I approximate $ \left(1+\frac{2}{4x-1}\right)^{x} $

I want to approximate $$ f(x) = \left(1+\frac{2}{4x-1}\right)^{x} $$ I begin with the Stirling Approximation $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^{n}$$ Raise both sides to the power of $$...
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What kind of equation am I trying to make? (factoral with addition type of thing)

So say I have a y = 12. Then I'd like to compute: (x*12)+(x*11)+(x*10)+(x*9)+(x*8)+(x*7)+(x*6)+(x*5)+(x*4)+(x*3)+(x*2)+(x*1) or ...
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2answers
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Ways to arrange seats where a certain group of people has to be together.

I just had this question on one of my quizzes and I have tried to do the question. So, we have 4 types of dogs on the day of the photoshoot, 1 golden retriever, 1 german shepherd, 4 chihuahuas and 4 ...
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1answer
73 views

Substituting large values of $n$ into Stirling’s formula, given the outcomes of other $n$ values

$$n! \approx \sqrt{2 \pi n} \; \left(\frac{n}{\mathrm e}\right)^{n},$$ in the sense that the percentage error $\to 0$ as $n \to \infty$. Show that the formula has an error of approximately $2.73\%$ ...
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12 views

Upper/lower bound for difference of two normalized gamma functions: $\frac{\Gamma(N, n)}{\Gamma(N)} - \frac{\Gamma(MN, Mn)}{\Gamma(MN)} = ?$

Is it possible to find the difference between the two normalized gamma functions with the following properties? What about an upper/lower bound? Note that $N, M > 1$ and $n > 0$. $\frac{\Gamma(N,...
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1answer
62 views

A closed form of $\frac{1}{2k+1}+\frac{2k}{(2k+1)(2k-1)}+\ldots+\frac{2k(2k-2)\cdots 4}{(2k+1)(2k-1)\cdots 3}$

Is there a closed form representation of the following sum? $$\frac{1}{2k+1}+\frac{2k}{(2k+1)(2k-1)}+\frac{2k(2k-2)}{(2k+1)(2k-1)(2k-3)}+\cdots+\frac{2k(2k-2)\cdots 6\cdot 4}{(2k+1)!!}.$$ Here $k$ is ...
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32 views

What factorial number to get a good approximation of the matrix exponential?

$$e^X = \sum_{k=0}^N{1 \over k!}X^k$$ Assume that $X \in \Re^{nxn}$ is random matrix. What number $N$ should I use to get a good accuracy compared if $N = \infty$? Is this possible to measure?
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1answer
33 views

sum of product of inverse factorials for all integers $m,n$ that sum to $k$

I'm wondering if there is a way to write the following in closed form in terms of the integer $k$ $$\sum_{\{m,n\}|m+n=k} \frac{1}{n!(m+1)!}$$ where, in words, I am summing up $\frac{1}{n!(m+1)!}$ for ...
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0answers
54 views

If $f(x) = (x+c)!$ and $g(x) = (ax+c)!$, how can I write $g(x)$ based on $f(x)$?

Consider $f(x) = (x+c)!$ and $g(x) = (ax+c)!$, where $a, x, c \geq 0$. Is there any way I can write $g(x)$ based on $f(x)$. I mean if I can write $g(x) = f(x)r(a, x)$, what would $r(a, x)$ be like? In ...
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3answers
59 views

how (a!)/(b!) = (b + 1)×(b + 2)×⋯×(a − 1)×a [closed]

I was solving a problem in which i need to figure out the prime factorization of $\frac{a!}{b!}$ and i did that by computing (a!) and then (b!) by looping ((1 to a) & (1 to b)) and then derived n ...
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356 views

How to evaluate the limit of multifactorial $\lim_{n\to 0} \sqrt[n]{n!!!!\cdots !}$

It is well known that $\displaystyle \lim_{n\to \infty}\sqrt[n]{n!}=\infty$, however, if we let $n\to 0$ we have different result with beautiful combination of $e$ and $\gamma$, that is $$\lim_{n\to ...
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2answers
91 views

Inequality involving factorial of sum

I noticed the following inequality involving factorials as a consequence of a statistics exercise: $$ (x_1+\cdots+x_n)!\leq n^{x_1+\cdots +x_n}\,x_1!\,\cdot\cdots\cdot\,x_n!\,, $$ where $x_1,\ldots,...
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2answers
64 views

Prove that (n!+1) is not divisible by any natural number between 2 and n. [closed]

I tried this question solving by mathematical induction. But no luck Is there is any easy way to prove that?
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16 views

A question about $ \ln (kc-1)! \sim \ln(c!(c-1)! \dots(c-k)!) + c \cdot\alpha$?

Question I was recently fiddling with some math that me wonder if there was nice asymptotic equation of the form $k<c$: $$ \ln (kc-1)! \sim \ln(c!(c-1)! \dots(c-k)!) + c \cdot\alpha$$ where $c$ is ...
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3answers
101 views

Is there a closed form of $\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$?

This may be an impossible problem. But I imagine it's worth asking still. What is the closed form of the sum: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$$ Perhaps there isn't a closed form. ...

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