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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2answers
1k views

Permutations excluding repeated characters

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the ...
5
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1answer
108 views

What is longer? Length of $2^n$ or number of $0$ in $n!$?

What is longer? Length of $2^n$ or number of $0$ at the end of $n!$ ? Assume that $n$ is really big, big number... Solution Look at terms of $2^k$: $$2,4,8,16,32,64,128,256,512,1024,2048,4096... $$ ...
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1answer
57 views

$n$-th derivative of $x^\alpha$ where $\alpha = m + 1/2$

It is well-known that, for any real $\alpha$ and nonnegative integer $n$ $$ \frac{d^n x^\alpha}{dx^n} = \alpha(\alpha-1)\cdots(\alpha - n + 1) x^{\alpha - n} $$ I just found out that the coefficient ...
31
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8answers
2k views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Let $m$ be a positive integer and $n$ a nonnegative integer. Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ \dfrac{\...
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3answers
47 views

Laplace approximation of Poisson posterior from MacKay

I am doing exercise 27.1 on Laplace's method from David MacKay's textbook, which is to make a Laplace approximation of a Poisson model with an improper prior: $$ p(x \mid \lambda) = \frac{e^{-\lambda}...
4
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1answer
77 views

Show that $(a!)^b b! \mid (ab)! $ [duplicate]

Show that if $a$ and $b$ are positive integers then $(a!)^b b! \mid (ab)! $ This is what I have done: The above statement is true for $a=1$ and any arbitrary value of $b$, and also for $b=1$ and ...
1
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3answers
67 views

For What Non-Negative Integer Values of $n$ is $n!\geq 3^n$

How could I solve for the $n$ in this instance using discrete methods or is this something that I have to do by hand/computer? I've seen this problem in inductive proofs but the base case is usually ...
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1answer
33 views

Simplifying an alternating sum of a product of factorials

For integers $a$ and $b$, I am curious how to simplify an expression of the form $$\sum_{k=1}^n (-1)^k (a+k)! (b+k)!$$ I assume there is some simplification using properties of gamma and beta ...
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0answers
24 views

What is the growth rate of a multivariable factorial divided by a multivariable factorial? [closed]

Consider the function $\frac{n!}{(n-2)!}$. The fraction simplifies to $n(n-1) = n^2 - n$, so the function has quadratic growth. Add a new variable into each factorial term: $\frac{(mn)!}{((m-2)(n-2))!...
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2answers
134 views

Probability of duplicated games in chess

I am just talking with a chess friend on Lichess about the probability to have duplicate games in chess, and some questions arose for me. Taken that a chess game always lasts $P$ (ply) half moves and ...
4
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1answer
254 views

Exponential and factorial that grow at exactly the same rate

I want to find a relation of the form $$ n^{n^a} = \Theta (n!) $$ I know and can reason fairly easily that $n^n$, where $a=1$, grows faster than $n!$, and $n^{\sqrt{n}}$, where $a=\frac{1}{2}$, ...
174
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6answers
8k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
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3answers
178 views

Solve $a!b!=a!+b!+c!$ where $a$, $b$ and $c$ are nonnegative integers.

My teacher in Math Team gave the following question to us. Solve $$a!b!=a!+b!+c!$$ where $a$, $b$ and $c$ are nonnegative integers. I found only one solution by trial and error and it is $(a,b,c)=(...
4
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1answer
49 views

Distribution of digits across all factorials

Show that the distribution of zeroes across all digits of all n! for $n\in \mathbb{N}$ converges to $ \frac{1}{6} $ and hence, $\frac{5}{54}$ for all other digits 1 through 9. In other words, let's ...
5
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1answer
403 views

What about irrationality of $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$? [closed]

I would like to learn more about the behavior of the factorial function or Gamma function, so I decided to compute $$\int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}\mathrm dx.$$ According to Wolfram alpha, ...
2
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2answers
82 views

How do I solve this combinatorics problem with conditions?

I have $N$ lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has $N-1$ ...
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1answer
42 views

How to prove this factorial equality

During my studies, I've come across this strange relation. $$\sum_{i=0}^{q-j-1}\left(\frac{\left(-1\right)^{i}}{i!\cdot\left(q-i-j-1\right)!\cdot\left(q+i\right)}\right)=\frac{\left(q-1\right)!}{\...
3
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1answer
22 views

Prove the given inequalities. [duplicate]

Prove the following inequalities,$$ (n!)^2 ≤ n^n(n)! <(2n)!$$ My attempt I proved one of the inequality using mathematical induction. To prove - $ (n!)^2 ≤ n^n(n)!$ For $ n = 1 $, LHS $≤$ ...
4
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2answers
103 views

$34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$

There was a number theory question that I have to do for homework. $34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$ I know $b=0$ because $10^7\big|34!$ only. But how can ...
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1answer
58 views

Prove that $n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$ [closed]

Prove that if $s_n=1^1+2^2+3^3+\cdots+n^n$ then $$n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$$ (it holds for $n$ larger than $2$). I want to prove ...
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4answers
125k 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 ...
5
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1answer
101 views

Four elevators stop on floors $1$ to $8$; a separate elevator stops on $1$ and $4$. Suppose you want to get from $1$ to $4$ …

I recently encountered an interesting math problem at the hospital. The hospital had two elevator banks. The first one had four elevators that went to any of the floors, 1 through 8. While the second ...
4
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2answers
98 views

How does $a^{a^n}$ compare with $n!$ asymptotically?

I am learning Asymptotic complexity of functions from CLRS. I know that exponentiation functions like $a^n$,$(a>0)$ are faster than $n!$ But what about $a^{a^n}$ vs $n!$ How do they compare? A ...
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5answers
12k views

Factorial number of digits

Is there any neat way to solve how many digits the number $20!$ have? I'm looking a solution which does not use computers, calculators nor log tables, just pen and paper.
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1answer
36 views

How do we obtain the following: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$

I saw the following equality in an informal proof: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$ I did not understand it and ...
3
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1answer
919 views

What is the theoretical upper bound of factorion numbers?

Recently I read about factorion numbers. I understood that there are only 4 factorion numbers, but what is the theoretical range in which they can be? Is it $[0, +\infty]$ or a smaller upper range? ...
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2answers
95 views

$x!=y^n$ for $x,y \neq 0,1$

A straightforward problem (find all integers such that $m!+3=n^2$) led me into thinking about the integers for which: $$x!=y^2$$ is true. I argued that other than the trivial case ($x!=1$) that this ...
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3answers
51 views

$100! $ in terms of $2^m Z$

Question I have encountered an question. If $$ 100! = 2^m Z $$ Where $Z\notin2\mathbb Z$ is an integer, find $m$ where $m \in \mathbb{ Z^+} $ My Attempt As $$100! = 2^{50} 50!$$ $[ 1×3×5×6. . . × ...
1
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2answers
89 views

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct?

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct, and $b$ and $d$ can not be both equal to $1$? For example, when $n=1319$, then $2^{11}-3^6=...
3
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2answers
124 views

Find all pairs $(k, n)$ of positive integers such that $k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$

Find all pairs $(k, n)$ of positive integers such that $$k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$$ I tried to solve this problem but only found one solution $(1,1)$. Please help me to ...
0
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1answer
992 views

Express Product of an Arithmetic Sequence in Terms of Factorials

How do you express the product of $5*8*11*14$ $*$ $...$ in terms of factorials that are functions of $n$, where $n$ stands for the number of terms in the product? Notice that the terms of the product ...
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2answers
32 views

Permutations after shuffling and drawing multiple cards from a deck

I'm trying to calculate the number of possible configurations of "starting game states" for a game that starts with shuffling a deck of 60 cards and drawing 7 cards. Some of the cards in the 60-card ...
3
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1answer
68 views

Prove that $~~~~~a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+…+a_m\right)! ~~~~~~~~~\forall ~~~a_1,a_2,…,a_m\in N$

Show that $$a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+...+a_m\right)! \forall a_1,a_2,...,a_m\in N$$ Case 1 $m=2$. We need to prove $$a_1!a_2!\mid (a_1+a_2)!$$ $a_1+a_2=1$ and $a_1+a_2=2$. It is ...
5
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1answer
159 views

Approximation of $x!$

I have discovered a new formula that approximates the factorial function. It is more accurate than Stirling’s approximation. How do I go about publishing it and receive the credit for it?
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1answer
49 views

Proving $\lim((n!)(\frac{e}{n})^n) = \infty$ using elementary method

Is there some easy way to show that $\lim_{n\to\infty} ((n!)(\frac{e}{n})^n) = \infty$ as $n \to \infty$? It looks like Wolfram alpha is using some kind of expansion method to numerically compute this?...
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1answer
41 views

solve for the lowest value of $k$ in ${n}\choose {k}$ $\geq$ $x$ , given $x$ and $n$

Let's suppose that $n = 10$ and $x = 500$. I want to find the smallest value of $k$ for which ${n}\choose {k}$ $\geq$ $x$ I can check for all values of $k$ from $1$ to $n/2$ and pick the least one ...
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2answers
36 views

how do I do I solve this equation [closed]

(n+1)! = 110(n-1)! I have searched online for a week now and my textbook does not explain how it got the answer: n=10.
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2answers
78 views

Query regarding a theorem in Number Theory

Please refer to Theorem $8$ in the attached picture. I do not understand the significance of the condition $(n+1)!+k>k$. Is it not always true since the inequation implies $(n+1)!>0$ which is ...
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1answer
54 views

Will someone please explain this equation from the picture

I am trying to understand the following equation: $$\sum_{k=0}^{\infty}\frac{a^k}{k!}\sum_{m=0}^{\infty}\frac{b^m}{m!} = \sum_{n=0}^{\infty}\frac{1}{n!}\sum_{k=0}^{n}\frac{n!}{k!(n-k)!}a^kb^{n-k} = \...
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3answers
110 views

On the determinant of a Toeplitz-Hessenberg matrix

I am having trouble proving that $$\det \begin{pmatrix} \dfrac{1}{1!} & 1 & 0 & 0 & \cdots & 0 \\ \dfrac{1}{2!} & \dfrac{1}{1!} & 1 & 0 & \cdots & 0 \\ \...
1
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1answer
22 views

Relation Gamma function and products

I have been thinking about a specific problem for quite some time. Imagine we have the following product, where $x \in \mathbb{N}$ and $a \in \mathbb{Z^+}$ then we now that the following holds: \begin{...
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1answer
46 views

Solving factorials! [closed]

If a!(n-a)!=b!(n-b)!, where a, b, n are naturals. Show that a=b or n=a+b. Also, n>a&b. I needed this prop. to prove one theorem. Help me! I don't need any rigorous proof. I will accept the answer, ...
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3answers
60 views

Limit of the sum of a factorial series

Is there any function $f$ over the positive integers such that $$\lim_{n\rightarrow\infty}\frac{\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}}{f(n)} = 1$$ and $$f(n)\not\equiv\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}...
21
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1answer
396 views

Computational results for the sequence $n!+{p_n}!+1$ are, well, very very unusual

Peter and I were discussing in a chat room and I thought that it would be nice to test the sequence $$n!+{p_n}!+1$$ for primality. Then I wrote Peter that I expect much of primes in this sequence, ...
0
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2answers
43 views

Designate set $ \lbrace n \in N : 2^{n-1} | n! \rbrace $

Actually I found out that if $n = 2^k$, $k\in Z_{+}$ we can say that it's true. Here's the proof: Let $n = 2^k$, $k\in Z_{+}$ Then $\nu _{2}(n!) = \nu _{2}((2^k)!) = \lfloor \frac{2^k}{2} \rfloor + \...
2
votes
3answers
61 views

Cannot find a calculator that can handle 220! x 40000

For a game I am playing, I am trying to calculate the cost of purchasing all the expansion area squares on a level. the first square costs 40,000, each next square costs an additional 40,000 more than ...
4
votes
3answers
908 views

What algorithms exist to quickly compute the inverse factorial?

I'm interested in algorithms to quickly compute the inverse factorial. I've noted that large factorials have a unique number of digits. How can I use this fact to quickly compute the factorial? Is ...
1
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1answer
49 views

What is the smallest integer $k$, for fixed $n$ (or vise versa), does $2^k! \geq 2^{n-k}$?

What is the smallest integer $k$, for fixed $n$ (or vise versa), does $2^k! \geq 2^{n-k}$? I tried to find $k$ for small $n$ so I could get some OEIS hit, but both sides just grow too fast for me to ...
2
votes
1answer
73 views

Functions whose Limit is the Factorial Function

I want to know examples of functions $f(n)$ whose limit is $n!$ Now, when I say "limit", I don't mean $$\lim_{n \to \infty}\frac{f(n)}{n!}=1$$ (I already know functions like that). I'm referring to ...
1
vote
1answer
44 views

Help with factorial simplification (n+1)*(n+1)!+(n+1)!-1 [closed]

Could someone show me the steps to get from $$(n+1)(n+1)!+(n+1)!-1$$ to $$(n+1)!\big((n+1)+1\big)-1$$ I'm having trouble understanding the simplification. Thank you!!!