Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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number of ways to choose pairs of nonadjacent people from $2k$ people sitting in a circle

The following is problem 19 in Chapter 2 from Richard Stanley's Enumerative Combinatorics, vol. 1 (2nd ed.): Suppose that $2k$ persons are sitting in a circle. In how many ways can they form pairs if ...
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0answers
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Generalized superfactorial notation.

I would like to know if there's a general shorthand notation for denoting the following product: $$\mathcal{P}=a!\times (a-k)!\times (a-2k)!\times\cdots\times (a-nk)!$$ where $a$ and $k$ both ...
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1answer
29 views

function of input size N, combination problem [duplicate]

Can someone please elaborate how from $(N+1)+N+(N-1)+(N-2)$ one can get $= 1/2(N+1)(N+2)$? also how to prove that: $(N-1)+(N-2)+...+3+2+1+0 = \frac{N(N-1)}{2} = {N \choose 2}$ ? Thank you!
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0answers
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Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
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2answers
197 views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
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2answers
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what is remainder when $(((3!)^{5!})^{7!})^{9!…}$ is divided by 11

$$(((3!)^{5!})^{7!})^{9!...}$$ when divided by 11 what will be the reminder? Hint is appreciated Sorry I do not know how to start this problem, so I have not shown my efforts!
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0answers
44 views

How to Find Number of Combinations

Here is the problem: In an experiment with eight trials involving the births of three children, what is the theoretical probability that you will get the distribution of: 0 girls-once 1 girl-three ...
3
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1answer
29 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
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2answers
102 views

$2\times 5 \times 8 \ldots \times (3n-1)=?$

Does anybody know if there is a closed form expression using factorials for the above product? I'm not seeing it but I feel like there must be. The recursive relationship corresponding to this ...
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2answers
53 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
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1answer
29 views

Intuitive explanation for this Gamma function identity

Wolfram Alpha says that this result is true: $$\frac{\Gamma(n+1)}{\Gamma(\frac{n}{2}+1)} = \frac{\Gamma(\frac{n}{2} +\frac{1}{2})}{\Gamma(\frac{1}{2})} \times 2^n$$ This implies a curious result for ...
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1answer
18 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
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6answers
68 views

Prove that $2^n(n!)^2 \leq (2n)!$

Prove that $2^n(n!)^2 \leq (2n)!$ One can also use the following result to prove the above: $2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}$. The above relation gives, $(2n)!=2^n n! ...
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3answers
101 views

Evaluating the factorial-related limit $\lim_{x \to \infty} (x + 1)!^{1 / (x + 1)} - x!^{1/x}$

I'm looking for the limit $$\lim_{x \to \infty} \left[[(x+1)!]^\frac{1}{1+x} - (x!)^\frac{1}{x}\right].$$ I've put the above in a computer program, and evaluated it at very high values of $x$ ...
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1answer
27 views

Calculate position in password search

I'm running a password cracker on my own password and I'm trying to calculate how long it will take. I know the rate the software is checking at and I also know the password. The password is $14$ ...
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2answers
47 views

Factorial Proof Problem

Suppose $m$ and $n$ are positive integers Prove $m!n! \lt (m+n)!$ I have something along the lines of: Since $1 \lt m+1$ and $2 \lt m+2$ etc.. then: $$n \lt m + n$$ So: $$n! \lt (m+n)!$$ I'm ...
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2answers
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Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
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2answers
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How many zeroes are there at the end of $36!^{36!}$?

Could you please tell me how many zeroes are there at the end of $36!$ to the power $36!$, i.e., $36!^{36!}$? I have been trying to find out. Read some reviews and answers related this but didn't ...
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2answers
475 views

Is this a meaningful approach to primes?

Motivation: I tend to be good at recognising patterns and I saw one with factorial: $$ n! = \prod_{i=1} p_i^{J(n,p_i)} $$ where $p_i$ is the $i$'th prime and $$ J(n,i)= \sum_{S=1}^\infty [n/i^S] $$ ...
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2answers
67 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
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4answers
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Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
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1answer
28 views

Induction Mathematics and Factorials

\usepackage{amsmath} Evaluate the sum $\sum_{k=1}^{n} {k\over (k+1)!}$ $\sum_{k=1}^{1} {1\over (1+1)!} = {1\over 2}$ $\sum_{k=1}^{2} {2\over (2+1)!} = {5\over 6}$ $\sum_{k=1}^{3} {3\over (3+1)!} ...
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2answers
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Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
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1answer
34 views

Evaulate a determinant involving factorials.

In a problem set given by a teacher, there is the following problem. If $a_n = \frac{1}{n!}$, evaluate $$ D_n = \begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0 & 0 & ...
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3answers
96 views

Why is Wolfram Alpha miscomputing this problem? [closed]

I was incorporating Wolfram Alpha into an API I am build, and to test it entered a few equations. One of the equations I entered was as follows. !6/(!3*!3) This ...
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3answers
481 views

How do you find the factorial of a decimal or negative number and what does it show us?

I know that you can find the factorial of positive integers where n!= n(n-1)...2 x 1. However, what if you want to find the factorial of a negative integer or a decimal? I tried to do it on my ...
6
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1answer
63 views

Finding all the zeroes in $100!$

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers) I know that to find the $0$s at the end we can use the greatest integer method. I was just ...
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1answer
106 views

Last nonzero digit of $2010!$ [closed]

I have to calculate the last nonzero digit of $2010!$ Till now I couldn't find any pattern.
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3answers
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Stirling Formula

Find the value of $\lambda$ for this question: $\dbinom{8n}{4n} \sim \lambda \dfrac{2^{8n}}{\sqrt{n}}$ as $n \to \infty$ I tried using Stirling. Any help appreciated.
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1answer
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Squeeze Theorem for Factorials

I have been having trouble with questions with factorials in Squeeze Theorem. This is the questions that I am struggling with: $\lim_{x\to \infty} {x^x\over(2x)!}$ What I have done so far: Lower ...
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1answer
69 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
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1answer
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Relationship between Factorial and Binomial coefficients

Over at this link, there is a claim that $(2n)! = n!n! {{2n} \choose {n}}$ - see Tom Boardman's answer, the second one down. I'm wondering why this is the case and if anyone can provide a proof. Is ...
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2answers
75 views

Need to show following equality

I want to show that the following equality holds for any integer i,m, and n.I could not figure out how to show it analytically. Could you please help me? $$ \sum _{j=0}^n ...
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3answers
67 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...
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2answers
61 views

Solving inequality equation involving sum of binomial coefficients

I have a function $f(k,\,i)$ involving binomial coefficients: $$f(k,\,i)\,=\left(\begin{matrix}k+i \\ k\end{matrix}\right)=\frac{(k+i)!}{k!\,i!}$$ And the following sum over this function (expansion ...
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4answers
157 views

What is the limit as k approaches infinity of $(k!)^{\frac{1}{k}}$ [duplicate]

What is the value of $$\lim_{k\to\infty}(k!)^{\frac{1}{k}}?$$ One of my students concluded the limit was infinity – which I tend to agree with, but was unable to show that was the limit. We ...
0
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1answer
37 views

Factorals with exponents. Is their a way?

I know of multiplication factorials with the 4! = 4*3*2*1 and I know of the addition with the nth triangle. I am busy deriving my own equation for something, and i am getting stuck on how to furthur ...
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3answers
90 views

Find $55! \bmod 61$

I am asked to find the smallest positive $x$ such that $x \equiv 55! \pmod{61}$. This invokes Wilson's theorem where $(p-1)! \equiv -1 \pmod p$. This means $60! \equiv -1 \pmod{61}$. But where to ...
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2answers
24 views

If $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$.

Using only precalculus knowledge, if $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$. (taken from Apostol's Calculus I, page 46) I don't ...
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2answers
47 views

In how many ways can you select one of the two but not both?

For this question: A committee of three boys and three girls is to be selected from a class of 14 boys and 17 girls. In how many ways can the committe be selected if: a.) Ana has to be on the ...
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0answers
19 views

What is the probability that a cluster of particles will contain some fraction of labeled particles, given total fraction of labeled particles?

Say you have a very large (but known) number of particles ($C$) and some known fraction of these particles are "labeled" ($F_L$). The particles spontaneously group into clusters of $n_c$ particles. ...
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2answers
103 views

Solve an equation involving factorial: $\frac{(n+1)!}{(n-2)!}=990$

For this equation: $$\frac{(n+1)!}{(n-2)!}=990$$ I need help with the working to the answer. Well I was stuck on the bit where I had ended up with: $$(n+1)n(n-1)=990$$ $$(n^2-1)n=990$$ ...
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2answers
36 views

Which rule is applied to define the operator precedence for factorial

Please apologize the question, I struggled with finding a good formulation in the first place: Looking at $\binom{2n}{k}$ it is very clear that for n,k integer and n>k we can solve it by calculating: ...
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1answer
91 views

Factorials: Simplifying $\frac{2017!+2014!}{2016!+2015!}$ to the nearest integer.

Compute $\dfrac{2017!+2014!}{2016!+2015!}$ to the nearest integer. My solution to the problem is 2016. Just wanted to check if it's correct. ...
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2answers
68 views

$\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$ [duplicate]

I know it's $\frac{\sqrt{\pi}}{2}$ but how can this be evaluated by hand? (Or can it not?) For quick reference: $$n!=\Gamma(n+1)$$ $$\Gamma(n)=(n-1)!$$ $$\Gamma(n)=\int_0^\infty x^{n-1} e^{-x}\,dx$$ ...
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0answers
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Proving non base specific factorial trailing 0 counting function

I have come up with the expression below to calculate the trailing zeros of $x!$ when represented in base $B$ $$\left\lfloor\sum^{\left\lfloor\log_n x \right\rfloor}_{r=1} ...
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2answers
53 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
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1answer
38 views

Summation with factorial

I want to understand how this step is performed. Can you tell me that how this value of Po is obtained from the first equation.! ...
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6answers
107 views

Is $\frac{n}{3}! = (\frac{1}{3})^n n!$

Is $$\frac{n}{3}! = (\frac{1}{3})^n n!$$ I thought I could take all the (1/3) out of the factorial, but wolfram alpha says this is false.
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4answers
85 views

$\lim_{ n\to \infty} \frac{n!}{n^n}$ via L'Hospital's rule

I just need to find this limit and I don't know how to use L'Hopital's rule in this case: $$\lim_{ n\to \infty} \frac{n!}{n^n}.$$ I apologize for the lack of formatting, I've never used the site ...