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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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 \infty}\...
0
votes
1answer
23 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, n_3)\...
1
vote
2answers
43 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 ...
1
vote
2answers
49 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
votes
1answer
54 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
votes
2answers
36 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
votes
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)}\ \sqrt{\pi}=\frac{\Gamma(2n)}{2\Gamma(n+1/2)}\sqrt{...
1
vote
3answers
57 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
votes
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
88 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 ...
1
vote
0answers
35 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
votes
1answer
89 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
votes
2answers
73 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
votes
1answer
40 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 $x,n\in \mathbb N$. ...
0
votes
3answers
40 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
68 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?
0
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^n-2^n)...
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
72 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
49 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
99 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
103 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
68 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
67 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
21 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
34 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
29 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
36 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
votes
4answers
119 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
34 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
49 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
151 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 2y}\...
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
63 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
24 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
66 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
48 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
40 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
1answer
33 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
84 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 ...
-8
votes
1answer
123 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.
1
vote
2answers
36 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
115 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
68 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 non-...
4
votes
2answers
65 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
111 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 $...