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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2
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1answer
42 views

Proving the summation of a double factorial infinite series.

$$ \sum_{n=0}^{\infty }\frac{(-1)^{n}((2n-1)!!)^2}{(2n)! (2^{2n})} = \frac{2}{\sqrt{5}} $$ I came across this summation through some other work, came across the solution as part of a function, but I ...
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1answer
29 views

Remainder theorem of factorials and powers

I took a competitive exam and there was this question If $7^n$ divides $68!$ Then what is the greatest value of $n$? Can someone tell me how to find the answer.
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1answer
38 views

Confusion regarding the scope of multiplying a factorial by a constant $30(3k)!$

I've stumbled upon this factorial, which is written exactly in this way: $$30(3k)!$$ As simple as this may seem, I honestly don't know how to interpret this. Does the factorial encompass the entire ...
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5answers
86 views

Find value of n if ${(n-1)! \over (n-3)!} = 30$

So here's my attempt : $${(n-1)(n-2)(n-3)! \over (n-3)!} = 30$$ $${(n-1)(n-2)} = 30$$ then $$n-1 = 30$$ or $$n-2 = 30$$ then $$n = 31$$ or $$n = 32$$ but my textbook says $$(n-1)(n-2) = 6 * 5$$ ...
9
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1answer
373 views

Scary looking limit with an elegant answer

This problem was posted half a year ago by Pierre Mounir on a Facebook group and until now it received no answers. Since most of his problems that I saw were amazing I can bet this one it's worth the ...
3
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0answers
28 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
4
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2answers
154 views

Does digit $6$ always lead to $\ 25921=161^2\ $?

Consider prime numbers with the property that the product of the factorials of the digits plus $1$ is a perfect square, for example the prime $$30241$$ leads to the square $$3!\cdot 0!\cdot 2!\cdot 4!\...
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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2answers
16 views

Subset of combinations in larger set

I am a biologist and not a real mathematician. Hence some of the answers featured here are sometimes too complicated. My question is: I have set of 8 genes named PBX1,ESX1,PIM1,HBB,HBG,BCL11A,KLF4,...
2
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1answer
48 views

What is the intermediate step in this negative binomial proof?

On Slide 47 of these slides, there is a formula stating $$ s_k(t) = - \sum_{j=0}^{k} s_j(0) \sum_{n=j}^{k} \frac{e^{-\lambda_n t} \prod_{m=j}^{k-1} \lambda_m }{\prod^{k}_{m=j, m \neq n} (\lambda_m - \...
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3answers
71 views

Determine $N$ of series $\sum_{n=1}^{N}\frac{n^n}{(2n+1)!}$ so that it differs from the actual sum by less than $\frac{1}{200}$

I can establish that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{n!}$$ But $$\sum_{n=1}^{+\infty}\frac{n^n}{n!}$$ diverges (by the ratio test). And even if it converged I wouldn't have ...
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3answers
81 views

Distribute 14 books to 2 people so that each has at least 3 books.

I've been trying to solve a problem, but the solution I got looks extrmely unlikely to be right. Perhaps some can point where I'm wrong. The problem is the following. We have $14$ different books. We ...
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2answers
167 views

In how many ways can 7 women, 10 men sit at table such that no woman sits besides another?

We have $7$ women and $10$ men; they sit at a table. I've been trying to solve how many ways can they sit excluding the case of women sitting next to each other. My reasoning was the following: I ...
0
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2answers
50 views

Given positive integers a and b, find values for a and b if a! * b! = a! + b! [closed]

Given positive integers a and b, find values for a and b if a! * b! = a! + b! I have no clue where to start and would appreciate some help, thanks!
0
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1answer
29 views

How to show that intermediate values of $\binom{n}{\lfloor n/2 \rfloor }$ are all integers that does not exceed $\binom{n}{\lfloor n/2 \rfloor }$

Let n be a positive integer, and assume that j is a positive integer not exceeding $n/2$. Show that in $n \cdot (n-1)\cdot (n-2)\dotsm(n-j+1)\,/\,j!$, if one alternates the multiplications and ...
2
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2answers
91 views

Why $ \lim_{n\rightarrow \infty} \frac{n!}{n^{k}(n-k)! } =1 $?

I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable. Consider the binomial ...
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1answer
22 views

Factoradic Operations

Recently I have been creating a program that uses Wilson's theorem to solve for prime numbers, but the main disadvantage of that system is that most computers have the 64 long limit, which comes out ...
6
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1answer
133 views

The behaviour of $\operatorname{Im}(!n)$

What's going on with the behaviour of the subfactorial's imaginary part? Background: Out of curiosity I tried to construct some recurrence relations using the Pochhammer symbol and out of those came ...
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0answers
31 views

How limitations affect the number of possible pandigital numbers

First, I understand that the definition of "pandigital" can vary, somewhat, so for the purposes of this question, here's the definition we'll use: "Positive and whole numbers that include each ...
4
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1answer
68 views

Solution of $n!=p+1 $ with $p$ is prime number?

One of my friend asked me to solve this equation $n!=p+1 $ with $p$ is prime number and n is positive integer , it's clear that for $p=2$ there is no solutions because : $n! < 3$ for $n=1$ , But ...
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3answers
55 views

Prove that $n$ factorial is larger than $n$ cube if $n$ is large enough

I try to solve it by Mathematical Induction. However I don't know how to prove that $(k+1)!>(k+1)^3$
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1answer
539 views

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
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3answers
56 views

How to prove that $\lim_{n \to \infty}\Bigl (\frac{n^n}{(2n)!} \Bigr )=0$

I'm asked to prove that $$\lim_{n \to \infty}\Bigl (\frac{n^n}{(2n)!} \Bigr )=0$$ I thought about using Stirling's approximation since $n \rightarrow\infty$ and then L'hopital in order to prove it ...
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3answers
78 views

Prove there are two positive constants for inequality

Prove, that there are two positive constants $C_{1}$ and $C_{2}$ such that $$C_{1}\sqrt{n}\left(\frac{n}{e}\right)^{n} < n! < C_{2}\sqrt{n}\left(\frac{n}{e}\right)^{n}$$ So I know there is ...
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3answers
38 views

For what values of $n$ is $10^8 < n! < 10^{12}$?

This question is from the introduction of a handbook on combinatorics. So far, the material covered contains: counting problems solved by using trees, Pascal's triangle,... definition of $n$ ...
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1answer
18 views

Manipulating a factorial involving multiplication and addition

I'm currently working on this problem which includes adding and multiplying (and probably factoring) factorials: $(k+1)!-1+(k+1)(k+1)!=(k+2)!-1$ I need to make the left hand side equal to the right ...
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1answer
51 views

Double Factorial expression in terms of regular factorials

I have the following double factorial expression: $$\frac{(x-2)!!(x-2k-1)!!}{(x-1)!!(x-2k)!!}$$ if x is an odd number I am trying to rewrite this without double factorials by using the following ...
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1answer
21 views

Some clarifications about Pascal's rule needed.

I have attached a screenshot of the proof of Pascal's rule and highlighted parts I don't understand. 1) why $(n-(k-1))$ on numerator is without factorial? I understand, that $(n-(k-1))$ is lesser ...
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1answer
67 views

How to solve this factorial equation?

How to solve this factorial equation? $$\frac{n!}{(n-6)!} = 350.418$$ or to give you the original equation: $$0.146 = \binom{n}{6} \times 0.45^6 \times 0.55^{n-6}$$ Sorry, but I've no idea about ...
2
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0answers
85 views

inequality for ratio of Gamma functions

Let $N,q$ be natural numbers. Find the best upper and lower bound (non-asymptotic) for $$ \frac{\Gamma(2q)}{\Gamma(q)}\frac{\Gamma(q+N/2)}{\Gamma(2(q+N/2))}=\frac{\prod_{k=1}^q(2q-(2k+1))}{\prod_{\...
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1answer
46 views

How can you further simplify a subtraction of two large factorials [closed]

100!-99! I am having a bit of difficulty with this question. Thanks in advance
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3answers
23 views

Ratio of large binomials in matlab

I need to compute the following ratio $$ \frac{n!}{j!(n-j)!}/ \frac{n!}{(n/2)!(n/2)!}$$ I've tried to do this using nchoosek which works finte until $n\approx1000$. But I need at least $n\approx 10000$...
2
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2answers
167 views

$\lim_{x \to \infty} {{(x!)^p}\over {x^x}}$

After some experimentation, I am pretty sure that $$\lim_{x \to \infty} {{(x!)^p}\over {x^x}} \to \infty $$ for $p \gt 1$, and that $$\lim_{x \to \infty} {{(x!)^p}\over {x^x}} \to 0 $$ for $p \le 1$. ...
0
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1answer
67 views

Quotient of factorials

Prove that $${(n^2)!\over(n!)^{n+1}}$$ 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 ...
0
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0answers
36 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 ...
0
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1answer
30 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
24 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 ...
0
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1answer
70 views

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

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
57 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
22 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
56 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 ...
3
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1answer
84 views

Maximising $\frac{x^n}{n!}$

Let $x$ be a number such that $x\gt 0$ and $x\in\mathbb{R}$. Is it true that the maximum value of the expression $$\frac{x^n}{n!}$$ occurs for $n\in\mathbb{N}$ where $n=\lceil x \rceil - 1$? If true, ...
2
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1answer
46 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.
0
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1answer
46 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
58 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
61 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 ...
-1
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2answers
66 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
136 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.
0
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1answer
38 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 ...
0
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3answers
90 views

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

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\...