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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54 views

Problem involving factorials (divisibility) [closed]

Show that, for every $n \in \Bbb N$, the following number is natural: $$\frac {(n!)!} {{n!}^{(n-1)!}}$$. I dont't know how to prove, as I tried to find a way including combinatorics.
6
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1answer
204 views

Factorials and the Mod function

I was just playing with the factorial and the modulo function. I just observed this interesting property. I was using a calculator $$13!\equiv 13\times 12\pmod{169}\\ 17!\equiv 17\times ...
0
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1answer
32 views

Trying to simplify an expression for an induction proof.

I got it down to $(k+2)!-1 + (k+1)((k+1)!)$ I am trying to get it to $(k+2)!-1$ but I guess I do not understand factorials enough to simplify this. I am also assuming I am doing the induction ...
1
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2answers
109 views

Calculate sum of infinite series by solving a differential equation

Calculate the sum of the infinite series $$\sum_{n=0}^{\infty}\frac{1}{(3n)!}$$ by solving an aptly chosen differential equation. I know that one can solve a differential equation by assuming that ...
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0answers
35 views

Using mathematical induction to prove P(n) [duplicate]

I have the statement $P(n)$: $2^n<(n+1)!$, for $n \geq 2$; $P(2)$: $2^2 < 3!$ which is true I.H P(k): $2^k<(k+1)!$ show that $P(k+1)$: $2^{k+1} <(k+2)!$ Here is my approach: ...
0
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2answers
59 views

Infinite Sum Calculation

How do we calculate this? $$ \sum_{n=1}^\infty \frac{n-1}{n!}$$ From my readings on previous posts here and through Google, I found that: $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ But then how do I ...
2
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3answers
206 views

Divergent sum of factorials

Is it possible to get an exact value of the sum (using divergent series summation methods) $$ \sum_{n=0}^\infty~ \frac{(n+k)!}{n!} \quad?$$ where $k$ is a positive integer. The only other divergent ...
0
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1answer
16 views

Simplifying an equation involving factorials.

I have found that the residue of the function $f(x) = \frac{1}{(1+x)^{n+1}}$ is $R= 2\pi i\frac{(n+1) \cdot (n+2) \cdot ... \cdot (2n)}{n! (2i)^{2n+1}}$. I am having trouble with showing that ...
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0answers
27 views

Divison in factoradic base

I'm trying to find am means of dividing two numbers in factoradic base. So far goolging seems to turn up nothing at all. Is there a better way of doing this than long-division? I'm hoping for ...
1
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1answer
37 views

How does factorials work in algebraic equations?

I was reading a text book and found these two lines and I have no clue how did step1 become step2. Please help me with this. Thanks Step 1: $$ \frac {5!}{(4-r)!} = \frac {6*5!}{(5-r+1)(5-r)(5-r-1)!} ...
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1answer
28 views

Evaluation of an expression

I have difficulties to evaluate this expression to the desired result. (It is a proof based on mathematical induction, left = right) $(k+1)!-1+(k+1)*(k+1)! = (k+2)!-1$
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0answers
23 views

What is $\prod_{k=1}^{(p-1)/2} (2k-1)^2 \bmod p$ where $p$ is an odd prime?

What is $\prod_{k=1}^{(p-1)/2} (2k-1)^2 \bmod p$ where $p$ is an odd prime? Note: Someone just asked this and it was deleted while I was working on it, so I am posting it with my answer. I get ...
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2answers
158 views

Is there a name for an 'incomplete' factorial $\frac{n!}{m!}$?

I noticed I was computing $${n! \over m!} ,$$ where $n > m$, inefficiently, as $$\frac{\prod_{k=1}^{n} k}{\prod_{k=1}^m k},$$ when many terms cancel out and I could just be calculating ...
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1answer
31 views

What function describes this problem of every possible breeding of a set of dogs?

If I have n dogs [a, b, c, ...], and I want to breed them in every possible combination (every possible binary tree made of ...
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5answers
94 views

How one can show $(r!)^s$ divides $(rs)!$?

I would like if anybody have suggestions to prove: if $n = rs$ with: $r > 0$ and $s > 0$ then $(r!)^s \mid n!$ suggestions?
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2answers
162 views

Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of k and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum ...
0
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2answers
54 views

Integral To Summation Problem

$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n ...
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1answer
44 views

Proof By Induction With Integration Problem

I am required to prove this formula by induction$$ \int x^k e^{\lambda x} = \frac{(-1)^{k+1}k!}{\lambda^{k+1}} + \sum_{i=0}^k \frac{(-1)^i k^\underline{i}}{\lambda^{i+1}}x^{k-i}e^{\lambda x}$$ where ...
1
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1answer
186 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor ...
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3answers
60 views

Is $n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)^i}{i!}}+…+\sum_{i=0}^{2}{\frac{(-1)^i}{i!}}\bigg]=(n-1)!$ true?

I am in the middle of doing a problem and has this sort of expression. I have a feeling that the following equality holds: $$n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! ...
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1answer
30 views

Negative factorial over another negative factorial [closed]

Suppose that you wanted to take the ratio of two negative factorials, such as: $\frac{(-n)!}{(-m)!}$ for positive integers $n,m>0$. I am pretty sure it is well defined (binomials with a negative ...
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1answer
94 views

Finding the $n^{th}$ derivative of $x^r$

I'm looking for a non-piecewise function -- $g(n,x)$ -- that satisfies this equation: $g(n,x)=\large\frac{d^{n}}{dx^{n}}x^{r}$ Where $n\in\Bbb{Z}$ and is the $n^{th}$ derivitive of $f(x)$ I ...
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2answers
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proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$

Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n =3$ For $n=3$, $1!+2!+3!=9=3^2$. I also feel that the word 'power' makes it a whole lot hard to prove. How do we prove this? ...
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1answer
97 views

Proof for this 'power triangle' [closed]

Is there a proof for this triangle Using factorial as a difference of powers? The first row is every consecutive integer raised to the power of n(5 here), but when you write the difference of them, ...
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1answer
32 views

Diophantine Factorial Equation [closed]

Prove that there exist pairwise distinct positive integers $a_0, a_1, a_2, \ldots, a_{1000}$ such that $a_0! = a_1!a_2! \cdots a_{1000}!.$
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3answers
74 views

Solving equations with factorials? [duplicate]

I looked on the internet but couldn't find anything relevant, so I was hoping you could help because I have no clue where to even start with how to solve this equations: x! = 6 Obviously trial and ...
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2answers
55 views

Limit of factorial quotient, what's wrong with my attempt? [duplicate]

$$\lim_{n \to \infty} \frac{n!}{n^{n}}$$ Attempt: $$n!=n\left( n-1 \right)\left( n-2 \right)...\left( n-\left( n-1 \right) \right)=n^{n}+...$$ $$\lim_{n \to \infty} \frac{n!}{n^{n}}=\lim_{n \to ...
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0answers
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What is the name of the identity: (2a)!=(a!*Γ(a-1/2) 4^a)/√π?

I just derived'(2a)!=(a!*Γ(a-1/2) 4^a)/√π'. My teacher has asked me to do some research over it. So my first question is. What is the name of this identity? Could it be a pi identity (if that is ...
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4answers
794 views

Prove that $n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!}$

Let $n$ be a positive integer. I conjectured that the following inequality is true \begin{equation} n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!} . \end{equation} Anyhow I could neither prove nor disprove it. I ...
4
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0answers
61 views

Find $a,b,c$ such that $a!\times b!\times c!=d!$ [duplicate]

I have to find $a,b,c \in \mathbb{N}$ such that-$a!\times b!\times c!=d!$ Answer given in my book is $3!\times5!\times7!=10!$(But it is written that other answers are also possible). What is a ...
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1answer
18 views

Is my intuition about this statistics problem sensible?

I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
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2answers
84 views

Verify the following identity algebraically

Verify the following identity algebraically (writing out the binomial coefficients as factorials).$${n \choose k}{k \choose m} = {n \choose m}{n-m \choose k-m}$$ So far, these are my steps: ...
0
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1answer
27 views

Probability of Bridge Hands Using Distributions

In a bridge deal, what is the probability that: a) West has five spades, two hearts, three diamonds, and three clubs? b) North and South have five spades, West has two spades, and East has one spade? ...
0
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1answer
38 views

One Dimensional Random Path Walker

Problem The Probabilities involving 3 equally possible moves in 1D line. Imagine a one-dimensional line with a "walker" in the middle position ($x=0$) Walker can make one of the following moves ...
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2answers
345 views

I am stuck on proving $\frac1{2!}+\frac2{3!}+\dots+\frac{n}{(n+1)!}=1-\frac1{(n+1)!}$ by induction, could anyone check my work?

I will skip the Base Case step. This is the questions. Use mathematical induction to prove that$$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$for all integers $n\ge 1$. ...
3
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2answers
78 views

Limit of a function including factorials

$$ \lim \frac{(2x)!}{(x! \cdot 2^x)^2} $$ How can I deal with problems including factorials as the same as this problem
3
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2answers
36 views

What value of c makes this true?

Since $\lim_{x \rightarrow \infty}\frac{(x)!}{x^{x}} = 0$ and $\lim_{x \rightarrow \infty}\frac{(2x)!}{x^{x}} = \infty$ Is there a value c (or range of values) where $\lim_{x \rightarrow ...
0
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1answer
22 views

Is it a correct equation for permutations with sets of indistinguishable objects?

C(n, r) = P(n, r)!/r! = n!/r!ㆍ(n-r)! I'll check if the right hand side of the above equation in Theorem 9.5.2 is correct by expanding the left hand side. $C(n, n_1)ㆍC(n-n_1, n_2)ㆍC(n-n_1-n_2, ...
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2answers
39 views

limit of fraction with factorials

I am trying to take the limit of the following fraction : $$ \lim_{N \to\infty} \frac { N !}{(N-r)!} $$ Attempts : I tried using the Stirling approximation $\ln(n!) =n \ln n - n $ but I figured it ...
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2answers
44 views

Excluding interval from N

I've recently been learning factorials in school. If there is an equation (in $\mathbb N$) with $(n-5)!$, I have to ensure that $n$ is not 1, 2, 3 or 4. I've been told that I should write domain: $D ...
2
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1answer
53 views

How to prove that $(p^2)!$ is divisible by $(p!)^{p+1}$?

For each prime $p$, find the greatest natural power of $p!$, which divides the number $(p^2)!$ ($n!=1 \cdot 2 \cdot ...\cdot n$) My work so far: 1) $p=2 \Rightarrow p!=2; (p!)^2=4!=24 \vdots 8=2^3$. ...
0
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2answers
35 views

Help required to compute logic of an answer.

How is the sum of combination series $${20 \choose 1} + {20\choose 2} + {20 \choose 3} +\cdots +{20 \choose 20} = 2^{20}?$$ No one told me or perhaps I missed the logic behind using so in my question. ...
0
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2answers
43 views

How to prove this gamma identity?

How to prove this? $$2^n \ \Gamma(n+\frac{1}{2})\ =\ 1.3.5...(2n-1)\ \sqrt{\pi}$$ I tried rewriting the right-hand side as $$\frac{(2n-1)!}{2(n-1/2)}\ ...
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3answers
56 views

What is $\int\limits_{0}^{1} \left[x(1-x)\right]^m \, dx$ ($m$ positive integer)?

I came across the following integral in my research: $$ \int\limits_{0}^{1} \left[x(1-x)\right]^m \, dx \qquad m\in\mathbb{N}^+ $$ According to my CAS (I use Matlab's Symbolic Toolbox), this ...
2
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1answer
14 views

Factorials and equivalency

I am not sure if this would be a proper title because I am a bit confused, but I was reading about proving Pascal's Triangle, and there was a proof on here I was following everything that was ...
2
votes
2answers
84 views

Does it make sense to multiply probabilities?

I got this interesting sum which seems to involve values of the derangement problem: $$ \sum _{n=0}^{\infty } \frac{1}{(2 n+2) (2 n)!}=\frac{e-1}{e}=1-\frac{1}{e},$$ where $1-\frac{1}{e}$ is the ...
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0answers
33 views

Limit of the spherical Bessel function of the second kind

I know that the limit for the spherical bessel function of the first kind when $x<<1$ is: $j_{n}(x<<1)=\frac{x^n}{(2n+1)!!}$ I can see this from the formula for $j_{n}(x)$ (taken from ...
4
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1answer
83 views

Is there a geometrical interpretation of this equality $2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$?

$$2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$$ How it can be seen in a plane? I have found many proofs with by induction but I wish to understand it geometrically.
0
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2answers
68 views

Upper Bound on ${n \choose r} $

$$ {n \choose r} \leq \frac{n^n}{r^r ~\cdot~ (n-r)^{(n-r)}} $$ I have a feeling that the above holds but I am not so sure how I go about proving it. Any insights?
2
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1answer
39 views

Minimize $x$ in factorial division

My question is that how can we find the smallest natural number, $n$, such that some other number, $x$, divides $n!$. What I mean is that what minimum $n$ such that $x\mid n!$ for $x,n\in \mathbb N$. ...