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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26
votes
11answers
9k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
2
votes
3answers
58 views

Constrictions on A.P with factorials.

There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression. Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
7
votes
3answers
250 views

Compact formula for $\sum_k k!$ [duplicate]

Is there any compact formula for: $$\sum_{k=0}^n k!$$ I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$ I've ...
2
votes
1answer
148 views

How to perform the summation/addition of binomial coefficients?

From my textbook: $$ \begin{align} \sum_{k=0}^n \binom {m+k}m &= \binom {m+n}m + \sum_{k=0}^{n-1} \binom {m+k}m\\\\\\ &= \binom {m+n}m + \binom {m+n}{m+1}\\\\\\ &= \binom {m+1+n}{m+1} ...
4
votes
4answers
5k views

What are the rules for factorial manipulation?

I know that $$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$ thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations ...
4
votes
2answers
134 views

Series involving factorials

How would one go about proving $$\int_{0}^1\frac{e^x-1}{x/2}\ dx=\sum_{n=0}^\infty\frac{1}{\binom{n+2}{2}}\frac{1}{n!}(0!+1!+2!+3!+...+n!)$$
7
votes
4answers
297 views

Combinatorial proof to $n! = (n-1)[(n-1)! + (n-2)!]$

It is for sure true that $n! = (n-1)[(n-1)! + (n-2)!]$ Since: $(n-1)(n-1)! + (n-1)(n-2)! = $ $(n-1)(n-1)! + (n-1)! =$ $ (n-1)!(n-1+1) = (n-1)!n = n! $ Today my friend told me that there is a ...
0
votes
1answer
223 views

Handling summations with two variables

If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$ $500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$, How would I handle the constant? If I ...
2
votes
2answers
184 views

How to prove this inequality with factorials

This is what I am trying to solve: $n!>n^{\frac {n}{2}}$ I tried with induction and somehow it doesn`t work this way, I tried with logarithms and also didn´t find a way. Is there an elementary(if ...
5
votes
1answer
356 views

Question about Ramanujan's proof of Bertrand's Postulate

I am reviewing Ramanujan's proof of Bertrand's Postulate which can be found here. At step #7, he writes: "But it is easy to see that..." $\log\Gamma(x) - 2\log\Gamma\left(\frac{1}{2}x + ...
3
votes
1answer
95 views

Understanding a very elementary property of factorials

I've seen this stated in a few places. If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} ...
3
votes
2answers
149 views

Factorials and Arithmetic Progression.

Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression. $n$ is a natural number I don't see any such examples(Except for $n=2$). And I don't see ...
3
votes
2answers
414 views

What does it mean to “have a multiplicative inverse of modulo 10!”?

Here's the question: What's the smallest integer > 1 that has a multiplicative inverse modulo 10! (that is, 10 factorial)? What does that mean? I understand that: We say that x is the ...
2
votes
3answers
148 views

Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$

How can we show that: $$3n< n!$$ whenever $n$ is an integer such that $n \geq 7$ ? I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
3
votes
3answers
107 views

Evaluate $\sum_{k=1}^{n}(k^2 \cdot (k+1)!)$

We have to evaluate the following: $$1^2 \cdot 2! + 2^2 \cdot 3! + \cdots + n^2 \cdot (n+1)! =\sum_{k=1}^{n} k^2 \cdot (k+1)!$$ Any hints ?
11
votes
1answer
128 views

Can this product be written so that symmetry is manifest?

Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression $$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$ apparently computes the ...
5
votes
2answers
109 views

Prove $((n+1)!)^n < 2!\cdot4!\cdots(2n)!$

so I know I need to prove this via induction, but I am somewhat stuck. Here is what I have does so far. Let $p(n) = (n+1)!^n \le 2!\cdot4!\cdot\ldots\cdot(2n)!$ $p(2) = 3!^2\le 2!\cdot4!$ Assume ...
1
vote
2answers
2k views

Summation of Series with Factorials

It is given that: $$v_n = n(n+1)(n+2)\;...\;(n+m)$$ $$and$$ $$u_n = (n+1)(n+2)\;...\;(n+m)$$ $i.$ Verify that: $$v_{n+1} - v_n = (m+1)(n+1)(n+2)\;...\;(n+m)$$ I started off by inspecting ...
0
votes
1answer
225 views

Finding modulus when all power of p are removed from N!

Given two integers $p$ and $N$. Let $m$ be number by $N!$ by max power of $p$ which divided $N!$. We have to find $m$ mod $p$. How to solve this?
12
votes
1answer
136 views

Solving $n!+m!+k^2=n!m!$ for positive integers $n,m,k$

I have been running in circles with this for a while now. It seems that the only solution is $(n,m,k)=(2,3,2)$ but I don't know how to prove it. Things I have noticed: WLOG $n\geq m$ we see that ...
19
votes
4answers
558 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
3
votes
0answers
40 views

Bound on Permutations [duplicate]

I am trying to prove the following inequality, $$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
3
votes
0answers
92 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
4
votes
3answers
221 views

Find all values of $n$ greater than or equal to 1 for which $n! + (n + 1)! + (n + 2)!$ is equal to a perfect square.

Not sure where to get started on this on. I started listing numbers for n starting at 1 but the numbers get very big very fast and I cannot find a pattern. Is there a better way of doing this or ...
7
votes
5answers
349 views

Find all solutions of the equation $x! + y! = z!$ [duplicate]

Not sure where to start with this one. Do we look at two cases where $x<y$ and where $x>y$ and then show that the smaller number will have the same values of the greater? What do you think?
5
votes
2answers
144 views

Is it significant that factorials have more trailing zeros as they get higher?

When I first learned about factorials in grade school I quickly became interested in the idea and did a lot of playing with them. I noticed, though, that as the factorials got higher and higher they ...
1
vote
4answers
2k views

Solving Equations with Factorials

I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given ...
5
votes
1answer
162 views

Generalization of the Factorial function

Is there any standard generalization of the Factorial function where the "skips" per multiplication is a parameter? For example, one generalization could be: $a(a-b)(a-2b)(a-3b)...1$ I tried to ...
5
votes
2answers
507 views

Ways to add up 10 numbers between 1 and 12 to get 70

I know this has something to do with factorials, and combinations and permutations. I've been puzzling over this for a little while, and I can't come up with an answer. My question is, How would one ...
2
votes
3answers
237 views

Prove that the combination formula can be reduced to…

Prove that: $$\frac{m!}{k!(m-k)!} = \frac{m}{k}\frac{m-1}{k-1}\cdots\frac{m-k+1}{1}$$ It's quite obvious when I write down some terms, but I just don't know how to make a rigorous proof. Any hints ...
0
votes
1answer
113 views

Simplifying a factorial containing only variables

I basically know how Im supposed to do this but I cant think of how to write it out on paper so someone else can follow what I did I need to find the limit of: $$\displaystyle\lim_{n \to \infty} ...
3
votes
4answers
10k views

Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do e consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? THanks.
2
votes
3answers
1k views

Simplification of fraction with factorials

I'm stuck on a simplification, used to prove $C(n - 1, r - 1) + C(n - 1, r) = C(n, r)$ Could somebody clarify the step(s) from: $\frac{(n - 1)!}{(r - 1)!(n - r)!} + \frac{(n - 1)!}{r!(n - r - 1)!}$ ...
0
votes
2answers
136 views

How to create equations to measure time spending in executing algorithms?

I made a program with two functions to calculate factorial. The first uses loops to made de calculations, and the second uses recursive calls to get the same result. The same program measures the ...
0
votes
1answer
40 views

Another inequality question

Can somebody in elementary way show that $(n!)^{2\over n+1}>n-1$ for only finitely many $n\in\mathbb N$? I need to prove this to be able to prove something else.
2
votes
3answers
248 views

Factorial expressed in terms of two other factorials

Can the factorial of $N$ always be expressed by the sum(addition and subtraction) or the product of two other factorials? Do there always exist integer $A$ and $B$ such that $N! = A! + B!$, or $N! = ...
0
votes
2answers
135 views

Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$
5
votes
1answer
221 views

A limit involves series and factorials

Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$
5
votes
1answer
108 views

What is the analytic continuation of a multifactorial?

The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials? I am particularly interested in the double factorial. All Google has ...
8
votes
4answers
607 views

Does $a!b!$ always divide $(a+b)!$

Hello the question is as stated above and is given to us in the context of group theory, specifically under the heading of isomorphism and products. I would write down what I have tried so far but I ...
2
votes
3answers
2k views

Sum of reciprocals of factorials

Could you help me count this sum: $$ \sum_{n=1}^{9} \frac{1}{n!} $$ I don't think I can use binomial coefficients.
7
votes
2answers
2k views

Number of zero digits in factorials

Here is a riddle someone has been asked in a job interview: How many zero digits are there in $100!$? Well, I found the first $24$ quite fast by counting how many times five divides $100!$ ($5$ ...
10
votes
1answer
332 views

How to solve $x!=5^x$?

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$. Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow ...
10
votes
2answers
243 views

Prove quotient of factorials is integral

If $n$ is an integer $\gt 0$, prove $$\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$$ is also an integer. I understand that a general approach is proving that the power of any prime factor is greater in the ...
1
vote
0answers
46 views

Approximation of factorial - Stirling formula [duplicate]

Possible Duplicate: Elementary central binomial coefficient estimates How can I prove that $$ \binom{n}{n/2} = \Theta\left(\frac{2^n}{\sqrt n}\right) $$ I tried with Stirlings ...
-4
votes
1answer
124 views

How many factors of $10$ in $100!$ [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How many factors of 10 are there in $100!$ (IIT Question)? Is it 26,25,24 or any other value Please tell how you have done ...
7
votes
1answer
161 views

Is n! mod p doable in sub O(n) time?

I ask because I can use Lucas Theorem to find n choose k mod p but don't know of an equivalent for permutations (n permute k mod p).
8
votes
3answers
428 views

Why is the double factorial $(-1)!! = 1$, by definition?

By definition, the double factorial $(-1)!! = 1$. How can this be rationalized?
5
votes
2answers
375 views

How to simplify this equality (factorials)?

This was in one of the examples of the textbook, but I couldn't figure out how they solved it. They say they multiply the left hand side by $\frac{n!}{n!}$ to get the right hand side: $$ \frac{2^n ...
1
vote
1answer
222 views

$n$ choose $k$ where $n$ is less than $k$

I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, ...