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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4
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1answer
43 views

How to calculate the fraction factorial without doing it manually?

We were asked to list the first $3$ terms of $$\frac1{\sqrt{4+3x}} = \frac12\left(1+\frac{3x}4\right)^{-\frac12}$$ which is straightforward and easy. Using the following with $n=-1/2$: $$(1+a)^n = {...
0
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0answers
9 views

Using Limits to extend the Factorials to Real numbers (and decimals)

Ref: Euler's limit formula for the factorial function I recently read about the Gamma function which works as a factorial operator for Natural Numbers. Most of the books even consider it as an ...
0
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1answer
27 views

Showing that the number of queues is uniquely expresssed as the product of $2010$ positive integers.

This problem came from a friend in preparation for a contest. There are $2011$ people in a queue lining up for a conference, and no two people have the same height. Bob is the $27$th tallest person ...
1
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0answers
56 views

Find the desired function or disprove its existence

Let $T(n,m)=\frac { n^2\cdot m\cdot f(n)}{n!}$. I need to find $f$ in terms of $n$, such that: $f$ is non decreasing function $f(n)\in\Omega(1)$ $\exists k>0.\ f(n)\in O(n^k)$ The following ...
3
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1answer
58 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 ...
5
votes
1answer
116 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 $2^k$: $$2,4,8,16,32,64,128,256,512,1024,2048,4096... $$ ...
5
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1answer
82 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
71 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 ...
0
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1answer
34 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 ...
1
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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))!...
0
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2answers
137 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 ...
2
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3answers
184 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)=(...
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
votes
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
votes
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 ...
-1
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1answer
60 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 ...
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 ...
4
votes
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 ...
0
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1answer
40 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 ...
2
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3answers
53 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. . . × ...
3
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2answers
130 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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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 ...
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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1answer
70 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?
2
votes
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?...
-2
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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.
0
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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 ...
0
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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 ...
1
vote
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} = \...
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{...
-6
votes
1answer
47 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, ...
0
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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)!}}...
1
vote
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 \\ \...
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 ...
0
votes
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
1answer
76 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
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 ...
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!!!
21
votes
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, ...
1
vote
3answers
77 views

limit of $\frac{\log(n^n)}{\log((2n)!)}$ as $n$ approaches $\infty$

I'm trying to solve this: limit of $\frac{\log(n^n)}{\log ((2n)!)}$ as $n$ approaches $\infty$ I know $\log(n^n)=n\log(n)$ using logarithms properties but what about $\log((2n)!)$? Idk how to solve ...
2
votes
0answers
28 views

Proof Involving Factors Of Arbitrarily Large Numbers [duplicate]

For prime $p$, show whether $$\prod_{p \geq 1} p^{\lfloor \frac{x}{p-1} \rfloor} \sim x!$$ as $x$ approaches infinity, and explain. I don’t know that it’s true, but I thought that it followed, if ...
4
votes
0answers
62 views

Primes $p$ which satisfy $p \mid \sum_{i=1}^{p-1} i!$

This question is inspired from @Mathphile's problem: The value$\sum_{i=1}^n i!$ where $n \in \mathbb{N}$, is only semiprime for $n=3,4$ One can easily solve this conjecture by knowing that $9 \mid ...
3
votes
0answers
67 views

Prove that $f(n)=n^{2007}-n!$ is an Injective Map

If $f: \mathbb{N} \to \mathbb{Z}$ defined as $f(n)=n^{2007}-n!$ Then Prove that it is an Injective function My try: According to the definition of Injective function: If $p,q \in \mathbb{N}$ and ...
2
votes
1answer
54 views

Show that $\frac{(m!)^{1/m}}{m/e}$ is a decreasing function of $m$

Show that $\dfrac{(m!)^{1/m}}{m/e}$ is a decreasing function of $m$. Here is my proof. I would like to see others, preferably simpler. I have shown in Proof explanation $\lim\limits_{n\to\infty}\...
0
votes
3answers
60 views

A series involving factorial

Series from $n=0$ to infinity of $n!/1000^n$ I know the limit of $n!$ is infinity and $1000^n$ is also infinity. In this regard, I really don't see how L'Hopital's rule can work in this case. How do ...
3
votes
2answers
163 views

Which one is bigger $100^{300}$ or $300!$?

How to find which one is bigger $100^{300}$ or $300!$ without using a calculator? I have tried it for whole 2 years but could not find it yet.
1
vote
2answers
79 views

Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
1
vote
1answer
35 views

Expressing the coefficients of $(1-x)^{1/4}$ using factorials

From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$ (1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n. $$ However, can I do the same ...