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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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20 views

Quotient of factorials

Prove that $\dfrac{(n^2)!}{(n!)^{n+1}}$ is an integer, where $n$ is a natural number greater than $5$. I know how the product of $r$ consecutive numbers is divisible by $r!$ Could we use it here? If ...
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0answers
18 views

Recurrence with non-fixed number of variables

Given the recurrence $\forall \{a_i\}\subseteq \mathbb N.\ \ T(\sum_i{a_i},m)\geq \sum_i {T(a_i, m-\prod_i (a_i!))}$ with $T(\Theta(1))=\Theta(1)$ How should I solve this? I tried to minimize the ...
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0answers
25 views

Factor of Kaprekar number

I'm trying to get the factor of Kaprekar Number, i.e: In range $1 \rightarrow 100$ there are $1, 9, 45, 55, 99$, so instead of checking all 100 number is Kaprekar Number or not, I'm trying to know ...
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1answer
24 views

Approximation $\ln \frac{(h+f)!}{(h-g)!}$ using Stirling’s when $f+g=o(h)$

Stirling’s approximation can be extended to a very well known inequality - $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \leq n! \leq e\sqrt{n}\left(\frac{n}{e}\right)^n$$ How can we use this to prove, ...
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0answers
16 views

Factorial function intersections

How would you find the intersection between a factorial function and exponential function? ie work out: $x! = 2^x$ Is it even possible ? As factorials only really work with integer values. When I ...
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2answers
67 views

Proof that $\sum\limits_{i=1}^\infty \frac{i}{(i+1)!} = 1$

I came across this result randomly and am quite sure it's right. Is there any way to prove it rigorously? The numerator always seems to be one less than the denominator. Thanks!
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0answers
31 views

Attempted algorithm to find which shortest permutation of a string out of “hard.”

Recently, I have been contemplating on how to find an unknown factorial. To find a particular string of text. Update- I removed pi. The formula Z is defined as L= length of string in character ...
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2answers
20 views

Show $\frac{ (n+k)!}{n! \sqrt{n+k}} \le ( f(n) )^k \sqrt{k!}$ for some function $f$

Let $k$ and $n$ be positive integers. Can we show the following inequality: \begin{align} \frac{ (n+k)!}{n! \sqrt{n+k}} \le ( f(n) )^k \sqrt{k!}, \end{align} where $f(n)$ is some funciton of $n$...
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0answers
52 views

Verification of $(-\frac{1}{2})!$

I was working on a proof for $(-\frac{1}{2})!$ and my issue was with converting my bounds from variable to variable. As you will see, I kept my bounds in terms of $t$ throughout the calculation, but ...
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0answers
27 views

Partial sum formula and indefinite integral of the following factorial/Lambert reciprocal function?

function is as follows: $$\frac 1{(2m)!(1-q^{2m})} $$ $$s.t. (m, q) \in \mathbb N$$ I ask this because of some difficulties encountered in a previous question I'd asked, linked here: Previous ...
2
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1answer
45 views

How to find summation of factorials

I got stuck at the following summation while solving another problem. $$\sum_{k=n}^N \frac{(k)!}{(k-n)!} $$ I expanded the summation but have no clue how to simplify it.
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1answer
38 views

Calculator giving weird answer when dividing factorial

I am using a TI-34 MultiView I was trying to divide the following 20!/(17!3!) The answer should be 1140 right? the numerator is 2.43*10^18 the denominator ...
8
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1answer
49 views

Similarity between $e^x$ power series and Gamma function integral?

The power series for $e^x$ is as follows. $$e^{x} =\sum ^{\infty }_{n=0}\frac{x^{n}}{n!}$$ If we define $n! = \Gamma(n+1)$, then we have $$n!=\int ^{\infty }_{0} x^{n} e^{-x} dx.$$ An extremely ...
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2answers
43 views

Permutation: How to arrange 12 people around a table for 7?

I want to understand how to arrange $12$ people around a circular table with $7$ chairs. We don't care about the overflow, those people can go to another table. I thought the way to solve the problem ...
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2answers
60 views

Show that $x! y! = z!$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$.) [closed]

Show that $$x! ·y! = z!$$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$) I am stuck on this problem. Within this section we are learning Congruence. So I know it involves ...
6
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1answer
133 views

How to solve $(x!)!+x!+x=x^{x!} $

How to solve this equation $$ (x!)!+x!+x=x^{x!} $$ The answer is $3$ . But I have no idea of how to solve it. Thanks for your time.
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1answer
33 views

Analytically continuing the product of the first $n!$ to negative numbers?

Analytically continuing the product of the first $n!$ I recently had the following idea to use the below identity: $$ (1!2! 3! \dots n!) (12^2 3^3 4^4 \dots n^n) = n!^{n+1}$$ If we focus on the ...
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3answers
69 views

How to calculate $\lim\limits_{x\to\infty} \frac{\log(x!)}{x\log(x)}$

How to calculate $\lim\limits_{x\to\infty} \frac{\log(x!)}{x\log(x)}$. Assume base $e$ (so $\ln)$. My attempt: $$\lim_{x\to\infty} \frac{\log(x!)}{x\log(x)}=\lim_{x\to\infty}\frac{\log(1\cdot 2\...
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2answers
62 views

What is the maximum value of $n$ if $4^n$ divides $1000!$ without a remainder?

If $1000!$ is divided by $4^n$ with a remainder 0, what is the highest possible value of $n$? I placed 2, 3, 4, etc value in $n$ but didn't found any possible $4^n$. Moreover I have seen that only $...
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2answers
37 views

Prove the following result

Prove that if $p$ is a prime number, then p divides $\binom{n}{p} − \lfloor\frac{n}{p}\rfloor$, for all $n > p$. (where the $\lfloor\frac{n}{p}\rfloor$ denotes the greatest integer less than or ...
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2answers
34 views

How do I solve this combinatorial proof involving factorial (n)_k?

Let $n$ and $k$ be positive integers with $n \ge k$. Give a combinatorial proof that $$n_k = (n-1)_k + k(n-1)_{k-1},$$ where $n_k$ is a falling factorial: $n_k$ = $n(n-1)(n-2)\ldots(n-k+1)$. I know ...
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1answer
18 views

Need Help with Some Advanced Integration By Parts Methods

Note: I am asking this question for someone to check my work for me. The problem started out with me finding z! which is equal to the $\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \...
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2answers
819 views

Find some digits of $17!$

$17!$ is equal to $$35568x428096y00$$ Both $x$ and $y$, are digits. Find $x$ and $y$. So, $$17!=2^{15}\times 3^6\times 5^3\times 7^2\times 11\times 13\times 17=(2^3\times 5^3)\times 2^{12}\times 3^...
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1answer
41 views

Lottery odds question

In this lottery 5 balls are chosen from 1-50. A friend offers me 5 to 1 if I can get one number correct from the 5 chosen, the order it comes up doesn't matter. Is this a good bet to take? To get ...
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2answers
47 views

Inequality with logarithmic function

Prove the following inequality for any $n ∈ [2;\infty)$: $$ \log_{n!}(\frac{n+1}{2})> \frac{1}{n} $$
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1answer
41 views

What are the steps to solve this problem? [closed]

If $\frac{1}{2n!}$ , $(n-2)!$, $(2-n)!$ are the side lengths of a triangle in cm., then what is the numerical value of the area of the triangle ?
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1answer
68 views

A subword in a word

Probability So I have been trying to solve this question in probability, but I don't seem to get the correct answer. I am not bad at probability and this seems to be easy one, but I'm just struggling ...
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0answers
27 views

Basic Question for limits and Taylor Expansion

I am trying to solve a question for limits. Is $1^x +2^x + 3^x + .....+ n^x$, a Taylor expansion for something? What is the Taylor expansion for $n! $? The question that I'm trying to solve is:- $$\...
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0answers
56 views

Bhargava's generalized factorials

Manjul Bhargava had generalized the factorial function in number-theoretic context in this paper. At the end of the paper, he mentions some interesting problems associated with the generalized ...
4
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1answer
38 views

Solutions to $1/K!+1/L!+1/M!=1/N!$

Is there more than one solution to $\frac{1}{K!}+\frac{1}{L!}+\frac{1}{M!}=\frac{1}{N!}$ where $K, L, M, N$ are all natural numbers? The one solution i came up with was to assume that $K=L=M$, ...
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1answer
32 views

Prove that given number is integer [duplicate]

I have bumped into one simple task which I am not able to prove: How to prove, that number $ \frac{1000!}{(100!)^{10}} $ is integer?
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0answers
25 views

Re-express this term as a binomial constant

How can I express the following constant $$\frac{n!}{q!k!r!(n-q-2k-r)!}$$ in terms of the Binomial constant or the falling factorial?
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2answers
147 views

Prove that $0!+1! + 2! + 3! + … + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three real number

Let $n$, $p$ and $r$ be three positive integers. Prove that for $n \geqslant 3, r>1$, $$\sum_{k = 0}^{n} k! \neq p^\text{r}$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preaparatory Question) I am not so ...
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1answer
25 views

Solutions for $n$? Use Stirling approximations if needed

$$(2n)! = a^{2n}$$ where $a \in \mathbb R$, and $n \in \mathbb N$. This is relevant because of a research question I'd asked and received an answer to by Sotiris here
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2answers
53 views

What's $\frac{(2n)!}{n!}$ equal to?

I chanced upon this problem: $\textbf{Show that} \hspace{0.2cm}\frac{(2n)!}{n!} = 2^n(1 \times 3 \times 5 \times ... \times (2n - 1)).$ I tried the following, and realised I was wrong! : $\frac{(...
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1answer
18 views

recursive relation on derangement of objects

Let $a_{n}$ represent the number of derangements of $n$ objects . If $a_{n+2}=p a_{n+1}+q a_{n}\;\forall n\in\mathbb{N}$ then what is $\displaystyle \frac{q}{p}$? What I have tried: I have used $$ ...
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1answer
32 views

Finding the last non-zero digit of $n!$ in $O(1)$

I saw a few approaches of finding the last non-zero digit using recurrence relation, CRT etc. I came up with a trivial $O(1)$ approach but didn't find it anywhere so asking it here. We can write $1\...
8
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1answer
96 views

$N! \pmod{P}$ (huge numbers)

What is the value of $2019! \pmod{7}$? I guess it's $0$? Because $$2019! = 2019\cdot2018\cdot2017\cdot ...\cdot7\cdot6\cdot...\cdot1$$ There's $7$ and also numbers that has $0$ remainder when divided ...
2
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2answers
49 views

can we perform modulo operator on a fraction on both of it's numerator and denominator?

I want to calculate nCr (mod $10^9+1)$.so for calculating nCr we have: $$nCr=\frac{n!}{r!(n-r)!}$$ so I want to know whether it is true that I perform modulo operator to numerator and denominator ...
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2answers
37 views

How to complete this proof involving factorials

Recently I came across the following identity, but if I try proving it with induction, then I get stuck. $$n! = \sum^n_{k=0}(-1)^{n-k}\binom{n}{k}(k+1)^n$$ While trying my induction step I get the ...
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3answers
48 views

A lower bound for de Polignac's formula

De Polignac's Formula has many uses, for example when calculating the number of trailing zeroes of $n!$ :$$\nu_5(n!)=\sum_{i\le\lfloor\log_5n\rfloor}\left\lfloor\frac n{5^i}\right\rfloor.$$ For the ...
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1answer
41 views

Does $N! = 2^m$ hold for any integer values of $N$ and $m$?

For any value of $N$, is it possible that the factorial of $N$ is equal to a power of 2?
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1answer
24 views

Proof for $\sum_{r=1}^{n}r(r!)=\sum_{r=1}^{n}[(r+1)!-r!]=(n+1)!-1$ [duplicate]

I came across the form $\sum_{r=1}^{n}r(r!)=\sum_{r=1}^{n}[(r+1)!-r!]=(n+1)!-1$ while solving a question in determinants. How do we get to the formula stated above?
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1answer
50 views

Storing numbers in an efficient way in computer.

I know that we can write a very large number such as 5040 in only 7!, and imagine I want to store this number in a binary file with the least number of bits. Saving 5040 takes 13 bits of space, while ...
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1answer
99 views

Evaluate the following limit: $\lim\limits_{ n\to\infty}\frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$

Evaluate $\lim_{n \to +\infty} \frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$. Please help with steps, Dont know how to break it down to cancel out terms.
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0answers
35 views

How to find the nearest factorial to a number

How can I find the nearest Factorial to a number? For example I know that the nearest factorial to 200 is 5! So how can I also ...
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1answer
58 views

Double-precision algorithm for inverse log gamma or log factorial?

Question in a nutshell: Can anyone point me to an algorithm for computing to double-precision floating-point (roughly 16 digits) the inverse of either log gamma or log factorial? In other words, if ...
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0answers
32 views

Counting number of occurrences of a number in a factorial

Consider that I want to count the number of times 360 occurs in 520! $360 = 2^3 \cdot 3^2 \cdot 5^1$ $520! = 1\cdot2\cdot3\cdot4\cdot\cdots$ As it can be noticed, $2$ occurs at least $3$ times ...
2
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1answer
22 views

Help with proving an equation factorial-time complexity

I've been recently asked by one of my friends to prove an equation but still, I'm confused how to get it started tho. log(n!)= θ(nlog(n)) Does anyone know how to help? I'll be very grateful if ...
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0answers
17 views

Reducing a large number to a smaller number using the Factorial Number System

Good day all, I have a large number 373335438 that I would want to reduce to a smaller number using the Facorial Number System here https://en.wikipedia.org/wiki/...