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

learn more… | top users | synonyms

1
vote
2answers
42 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 ...
0
votes
0answers
18 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. ...
0
votes
2answers
96 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: $$(n+1)n(n-1)=990$$ $$(n^2-1)n=990$$ ...
2
votes
2answers
30 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: ...
6
votes
1answer
87 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. ...
3
votes
2answers
66 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: $$n!=\Gamma(n+1)$$ $$\Gamma(n)=(n-1)!$$ $$\Gamma(n)=\int_0^\infty x^{n-1} e^{-x}\,dx$$ ...
0
votes
0answers
9 views

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} ...
1
vote
2answers
50 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 ...
0
votes
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.! ...
1
vote
6answers
104 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.
1
vote
4answers
84 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 ...
0
votes
2answers
36 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
0
votes
1answer
24 views

Proving two equations containing ceiling and floor function to be equal

The problem is to show that if $n$ is a positive integer, then $$1 = C(n, 0) < C(n, 1) < ... < C(n, \left \lfloor{ n/2} \right \rfloor) = C(n, \left \lceil {n/2} \right \rceil) > ... > ...
2
votes
1answer
53 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
2
votes
8answers
128 views

How can $0!=1$ if the definition of factorial is $n!=n\times (n-1)!$ [duplicate]

Its a pretty basic question. If the definition of factorial is $n!= n\times(n-1)!$, then how can $0!=1$ since if we feed $0$ into the equation we get $0!=0\times (-1)!$? This comes after a ...
2
votes
2answers
72 views

Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$ n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this ...
1
vote
3answers
41 views

How to calculate $\lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$

I need to calculate limit number 1, and I don't understand how to get out the factors. $$ (1) \lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$$ $$ (2) \lim_{k \to \infty} \frac{(k)!}{(k+1)!}$$ When I ...
0
votes
1answer
33 views

function to approximate $x!$ without factorial

I am looking for a function $f(x)$ such that $f(x)\approx x!$, but (obviously) the function of x does not use factorial, eg a polynomial or exponential function. it does not have to be precise, just ...
0
votes
0answers
25 views

Coefficients for the falling factorial

Hello fellow mathematicians, I am trying to find a generating function, or at least find some useful property from the coefficients of the falling factorial. Let $(x)_n$ denote a falling factorial, ...
2
votes
1answer
41 views

integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$

Total no. of integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$ $\bf{My\; Try::}$ Let $w=\max\left\{x,y\right\}$. Then $w<z$. So we can write $w\leq (z-1)$ So ...
2
votes
1answer
35 views

Proof for 1/k! using n choose k as n approaches infinity and its relation to the gamma function

Prove that $\lim_{ n \to \infty }\binom{n}{k}(1/n)^k =\frac{1}{k!}$ How is this related to the gamma function?
4
votes
1answer
63 views

Combinatorial interpretation of double factorial.

Using some basic algebra (and proved afterwards using induction), I found that: $$ 1 \cdot 3 \cdot ... (2n-1) = \frac{(2n)!}{2^n \cdot n!}$$ After a bit of research, I found out that this is known ...
17
votes
4answers
462 views

Interpreting $n!$ as the volume of a $1 \times 2 \cdots \times n$ box

Q. Are there relationships or proofs that are illuminated by viewing $n!$ as the volume of a $1 \times 2 \cdots \times n$ box in $n$-dimensions? I cannot think of any, but perhaps they ...
2
votes
2answers
48 views

How can I express the ration of double factorials $\frac{(2n+1)!!}{(2n)!!}$ as a single factorial?

How can I change the double factorial of $$\frac{(2n+1)!!}{(2n)!!}$$ to single factorial?
0
votes
2answers
43 views

Show that $n! < (n/2)^n$ for all large enough $n$ in as elementary a way as possible

Show that $n! < (n/2)^n$ for large enough $n$ in as elementary a way as possible. Using Stirling's formula is not allowed. Of, course, what is true, is that $n! < (n/c)^n$ for any $c < e$ ...
3
votes
1answer
37 views

Expressing $\sum_{k=1}^{n}\frac{1}{(k+2)k!}$ in terms of $n$.

How would I express $$\sum_{k=1}^{n}\frac{1}{(k+2)k!}$$ in terms of $n$? An attempt of mine is $$\sum_{k=1}^{n}\frac{1}{(k+2)k!} = \sum_{k=1}^{n}\frac{1}{(k+1)! + k!},$$ which is not useful for ...
2
votes
1answer
91 views

Factorials…How do they do it?

So I've been recently arguing with my teacher about factorials. My teacher says that factorials can only be calculated for integers, because the definition of factorials is as follows: the product ...
2
votes
2answers
64 views

Summation over a product of binomial coefficients

Question: I can't figure out why the following equality is true $\sum_\limits{k=a-b-c}^{d} (-1)^k \binom{d}{k}\binom{k+b+c}{a} = (-1)^d \binom{b+c}{a-d} $ How can this be shown? (In the book it just ...
6
votes
2answers
100 views

Sum of factorial fractions

Find the sum $$\sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!}$$ I tried making something like a geometric series but couldn't. Then I couldn't think ...
2
votes
1answer
67 views

Floor function of a factorial

Compute $$\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.$$ How can I start with the problem? I thought of dividing by some number, but then I thought that some small numbers when added ...
-2
votes
1answer
46 views

Inverse question of trailing zeros [duplicate]

$(5n)!$ has $2014$ trailing zeros. What is $n$?
-5
votes
2answers
45 views

Aptitude test question… [closed]

How does 4(4!)=24? (and not 96)
4
votes
0answers
65 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
5
votes
4answers
359 views

Easy Double Sums Question: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(m+n)!}$

How to calculate $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(m+n)!} $ ? I don't know how to approach it . Please help :) P.S.I am new to Double Sums and am not able to find any good sources ...
1
vote
4answers
51 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
2
votes
1answer
25 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
0
votes
4answers
33 views

Show $(2n+2)!\geq(n+2)(n+2)!$, $\forall n\in\mathbb{N}$.

It's the last step in a proof, and I just need to show that $$(2n+2)!\geq(n+2)(n+2)!$$ $\forall n\in\mathbb{N}$. I can't seem to do it though, any thoughts?
2
votes
2answers
44 views

Proof of factorial inequality concerning fractions

I'm having trouble with a proof, with the case $n>2$. THEOREM: For every natural number $n∈N$ where $n≠2$, $∑_{i=1}^ni≤n!$ Let us simplify the statement. ...
3
votes
4answers
107 views

Determine if this series $\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$ converges or diverges, and prove your answer?

Determine if this series $$\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$$ converges or diverges, and prove your answer? I've been able to prove similar problems, but I'm confused now that there's a ...
0
votes
1answer
39 views

Calculating p-adic valuation $v_p(n)$, using basic properties

Calculating p-adic valuation $v_p(n)$ I'm not confident with the properties of $v_p(n)$ Where $v_p(n) = $ the biggest integer $e$ such that $p^e$ divides $n$, if $n\not=0$, and $+\infty$ if $n=0$. ...
2
votes
3answers
42 views

Lower bound for the falling factorial $(2n)_{n}$

I'm looking for a lower bound for the falling factorial $$(2n)_{n}:= \frac{(2n)!}{n!}$$ I saw on Wikipedia that $n! > \sqrt{2{\pi}n}(\frac{n}{e})^n$ . So I assume that a possible lower bound ...
0
votes
2answers
65 views

Find the number of trailing zeroes. [duplicate]

Find the number of trailing zeroes. $k=1^1\times 2^2\times 3^3\times \cdots \times100^{100}$ It usually involves calculating number of $5$'s in $5^5\times 10^{10}\times 15^{15}\times \cdots\times ...
1
vote
2answers
70 views

Formula for factorial?

I need an equation that defines factorial without using factorial, that also works for $0$. I have seen factorial defined like this: $$n! = 1\cdot2\cdot3\cdot4\cdots n$$ But if we plug $0$ into that, ...
0
votes
2answers
257 views

Why is it defined that $(-1)!!=1$?

Why is it defined that $(-1)!!$ equal to $1$, where $!!$ is the double factorial? I've only seen it defined that $(-1)!!=1$, but I don't see why it should be so.
0
votes
2answers
14 views

What's the difference between derangements and partial derangements?

What's the difference between derangements and partial derangements? I know that Derangements are essentially subfactorials; could anyone explain the difference? I came across this in some local ...
2
votes
1answer
42 views

$\frac{N!}{(N-n)!}$ when $n<<N$

I need to show that for $n<<N$ then $\frac{N!}{(N-n)!} \approx N^{n} $ I can see that $\frac{N!}{(N-n)!} = (N)(N-1)...(N-(n-1))$ and intuitively its clear but I am unable to show rigorously. ...
1
vote
2answers
25 views

Combination vs permutation

A teacher has $5$ books to distribute to some of $20$ children in her class. How many ways are there for her to distribute the books if the books are all the same and no child gets more than one? ...
4
votes
1answer
85 views

Finding missing digits in factorials

14!=871a82b1200 without working out 14!, find a and b I think it has something to do with maths rules regarding 9 or 3 (the digits adding up to either of those numbers) but not entirely sure!
9
votes
2answers
498 views

when is $n!+10$ a perfect square?

When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
2
votes
0answers
230 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...