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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0
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3answers
39 views

How can I test the convergence of the following sequence with odd products over even ones? [closed]

The sequence is: $$a_n = \frac {2^{2n} \cdot1\cdot3\cdot5\cdot...\cdot(2n+1)} {(2n!)\cdot2\cdot4\cdot6\cdot...\cdot(2n)} $$
1
vote
5answers
67 views

Prove that $\frac{(n+1)!}{(n-1)!}=n(n+1)$

Simplify $$\frac{(n+1)!}{(n-1)!}$$ My book shows the answer as $n(n+1)$. I don't know how does it come up? I have tried: $$\frac{(n+1)!}{(n-1)!}=\frac{(n+1)n!}{(n-1)(n-2)...n!}$$ I have been ...
12
votes
4answers
186 views

Prove that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$ [closed]

Using a graphing calculator, it seems that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$. How can this be proven?
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votes
1answer
36 views

Calculate chosen element of this Number Pyramid

Introduction Lets choose a natural number $n$ and generate a sequence $ k_1, k_2, k_3, ... k_n $ where each $k_n$ is replaced using $ f(k_a)= (a+1)^n - a^n $ The sequence then is $ (2^n-1), ...
3
votes
1answer
62 views

Finding the solutions of $n! \ge n^a$

Let $a \in \mathbb{N}, a \ge 2$ be a fixed natural number. Consider the inequality: $$n! \ge n^a$$ It can be proven that this inequality is true for sufficiently large values of $n$, but how can we ...
2
votes
2answers
70 views

Simplify and find $\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}$

So I was calculating $$\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}$$ and couldn't solve it, so I saw the answer sheet and it said that the limit was $0$, I checked the process and they simplified the ...
0
votes
2answers
43 views

Rectangular Table Arrangement

a) In how many ways can 13 people be seated on one side of a rectangular table if Doug refuses to sit next to Gordon? I have two different ideas- Idea 1) There are two options: either Doug is at ...
3
votes
2answers
93 views

Expressing Factorials with Binomial Coefficients

Expression I have somehow stumbled upon this expression (I believe I have proved it, but that is not important right now), which I have tried to simplify by writing it like something like this (I ...
3
votes
1answer
102 views

Prove that any prime factor of $ ( x!+1)$ is larger than$ x$.

I want to prove the statement "Any prime factor of $x!+1$ is larger than $x$." Any slight hint will be ok.
2
votes
4answers
67 views

Does the sequence $a^n/n!$ converge?

The sequence when plotted converges to zero because a factorial grows faster than the numerator, but I can not prove that this sequence actually converges.
1
vote
1answer
65 views

Find the value of n if:

$$\sum_{k=0}^n (k^{2}+k+1) k! = (2007).2007!$$ How to approach this problem? In need of ideas. Thank you.
0
votes
1answer
19 views

Proper factorial answer

I'm new to factorial equations and wanted to see if I got it down properly. I have 35 possible numbers, so N = 35. I have 36 possible length, so R = 36. This is a combinations where numbers can be ...
1
vote
0answers
33 views

Is there a general formula to the: sum n! [duplicate]

For example as the: sum n has the general formula >>> n(n+1)/2 Is there a formula for the n!
1
vote
3answers
27 views

Proving identity with factorials: $K\cdot(K - 1) \cdot(K - 2) \cdot \dots\cdot (K - N + 1) = K!/(K-N)!$

I was looking at some formulas in one of my courses and I see the following equality: $$K\cdot(K - 1)\cdot(K - 2)\cdots(K - N + 1) = \frac{K!}{(K-N)!}$$ Can't see a way to figure out how these are ...
0
votes
2answers
34 views

Simplification of Log Factorial Expression

I would like to find a simplification of the expression \begin{equation} \log{\frac{(x+y+z)!}{x! y! z!}} \end{equation} that is linear with respect to $x, y$, and $z$. Does such an expression exist? ...
9
votes
5answers
1k views

Factorial Calculation for Non-Integers?

I was playing with numbers on calculator and to my amaze i could see that calculator calculated $(4.5)!$ or any real numbers but factorial is defined for integers how is this done any advanced ...
5
votes
4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
4
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4answers
117 views

How to calculate $\sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i}$

how can we calculate this?$$ \sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i} $$ It is alternating and contains the Binomial coefficients which are given in terms of factorials as, $$ ...
0
votes
0answers
31 views

One-sided limit with factorial/Gamma function

I was wondering if the following limit could be evaluated: $$\frac{\lim_{x\to0^+}\Gamma(x)}{\lim_{x\to0^-}|\Gamma(x)|}$$ I wondered if we could take the limit as $x$ approached $-1$ and compare the ...
3
votes
1answer
48 views

Is the Gamma function defined for complex numbers like so?

I know $\Gamma (x)$ can be defined as... $$\Gamma (x) = (x-1)!$$ And as... $$\Gamma (x) = \int_{0}^{\infty} u^{x-1} e^{-u} \space du$$ But assuming we had a Complex variable $z$ and had the ...
6
votes
1answer
143 views

Orthogonality relation as double sum of products of binomial coefficients

I have stumbled upon the following sum over $x,y$ for non-negative integers $\kappa,\lambda$: $$ \sum_{x=0}^{\kappa}\sum_{y=0}^{\lambda}\left(-\right)^{x+y}{2\kappa \choose 2x}{2\lambda \choose ...
2
votes
3answers
37 views

Different seating arrangements on a 6 seater [closed]

Six actors sit in a row to have their photographs taken. Romeo and Juliet insist on sitting next to each other. Caesar refuses to sit next to Brutus. Falsta and Puck don't mind where they sit. How ...
3
votes
1answer
58 views

Can you find it? or Can you prove it? [duplicate]

$x!+1=y^2$, I found 3 solutions. They are $(4,5),(5,11),(7,71)$. Is there a $4$th solution?If not can you prove it?
2
votes
1answer
23 views

Binomial Distribution and Proof Relating to Factorials

I am studying probability and statistics at my university but haven't had a solid math course in awhile(mostly forget algebra dealing with factorials)thus I am stuck with the following proof. ...
0
votes
3answers
65 views

How many ways to write sum as $k$ restricted integers

If we have $3$ positive integers given as \begin{align*} 0 & \leq a \leq 3\\ 0 & \leq b \leq 3\\ 0 & \leq c \leq 2 \end{align*} and given the sum $$ a+b+c = 5$$ how many ways we can ...
1
vote
2answers
46 views

Representing geometric series as sum of binomial coefficients

I was learning generating functions, where faced following: $$(1-x)^{-n}=\sum_{i=0}^\infty\binom{n+i-1}{i}x^i$$ I didn't get how is this arrived. I tried out to prove this to me. But failed. ...
0
votes
0answers
33 views

How many ways to pile up boxes in a direction

Lets assume that there are columns with spesific limits, and there are boxes on these columns. We need to find all the possible ways(positions) from the original layout to the layout that completely ...
0
votes
0answers
20 views

Finding maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
3
votes
2answers
75 views

How many ways we can choose items from different boxes

I searched through the internet but couldn't find my answer, which can either be a very simple or a hard one. Assume there are $3$ boxes, which carry, respectively, $1$, $4$, $2$ items. My question ...
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votes
1answer
103 views

What is product of $1!\cdot2!\cdot 3!\cdot…\cdot n!$ [closed]

Suppose that $F$ is the required function. I need the value of this function till $n$ natural numbers with a direct mathematical expression.
0
votes
2answers
32 views

Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...
0
votes
0answers
31 views

Is there a proof that $(n-2x)! \times n^{2x-1} > n!$ (where $x$ is a function related to the prime counting function

Is it possible to prove the following? Let $\pi$ be the prime counting function and $A(n)=\pi(2n)-\pi(n)$ $(n-2A(n))! \times n^{2A(n)-1} > n!$
2
votes
2answers
107 views

Is there any better way to find n! without just multiplying all the numbers till n? [duplicate]

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till ...
0
votes
2answers
66 views

Prove or disprove: For non-negative integers $m$ and $n$, $m!n! = (mn)!$

I have rewritten the question as "If $m$ and $n$ are non-negative integers, then $m!n!$ = $(mn)!$" Here is my current attempt. I am not sure if I am on the right path. Proof. Let $m$ and $n$ be ...
3
votes
2answers
53 views

Generalizing Legendre's Theorem on prime factorization of factorials

From Legendre's Theorem we know $$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$. Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for ...
4
votes
2answers
106 views

Last digit of $235!^{69}$

Problem What is the last digit of $235!^{69}$? It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back. My initial thought goes to the last digit ...
3
votes
1answer
49 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
2
votes
2answers
55 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
0
votes
2answers
41 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
5
votes
2answers
150 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
0
votes
2answers
77 views

The method of solving for a factor of $90!$ [duplicate]

If $90! = (90)(89)(88)...(2)(1)$, then what is the exponent of the highest power of $2$ which will divide $90!$ ? How would I apply one of the easiest method from Here? I need help on applying ...
20
votes
1answer
493 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
5
votes
5answers
179 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
2
votes
0answers
32 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
116
votes
5answers
5k 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 ...
1
vote
4answers
66 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
1
vote
2answers
35 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
5
votes
3answers
537 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
2
votes
4answers
79 views

Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
0
votes
2answers
33 views

Factorial with names

Ok so, I have had an argument with my teacher over 1 quiz question that was marked wrong in my data management class. Question. Determine the number of ways that 12 members of the boys' baseball team ...