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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10
votes
1answer
141 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
113 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
252 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
141 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
624 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
98 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
262 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
392 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
195 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 ...
2
votes
4answers
4k 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
232 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
861 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
302 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
134 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} ...
6
votes
4answers
18k 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
2k 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
140 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
41 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
357 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! = ...
1
vote
2answers
262 views

Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$ [closed]

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$
5
votes
1answer
255 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
126 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 ...
9
votes
4answers
623 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
3k 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.
9
votes
2answers
3k 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
347 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
273 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
53 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
145 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
167 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
550 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
444 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
323 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 $_0C_1$, $_0C_2$, ...
3
votes
1answer
942 views

Calculating a large factorial division on pen and paper

Context: In January I will be taking a concurrent programming examination, part of which will involve calculating the number of interleavings based on a formula which divides two sets of factorials. ...
2
votes
3answers
217 views

Convergence of Sequence with factorial

I want to show that $$ a_n = \frac{3^n}{n!} $$ converges to zero. I tried Stirlings formulae, by it the fraction becomes $$ \frac{3^n}{\sqrt{2\pi n} (n^n/e^n)} $$ which equals $$ ...
0
votes
2answers
157 views

Prove by induction that $n! > n^2$

How does one prove by induction that $n! > n^2$ for $n \geq 4$
7
votes
6answers
325 views

simplify summation of factorial (random walk)

I suspect that the expression $$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$ simplifies to $$\frac{2^N}{(N-1)!}$$ But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
2
votes
3answers
273 views

Algebraic manipulation of binomial theorem

Prove, by algebraic manipulation, that: \[ {{2n} \choose {n}} + {{2n} \choose {n+1}}={1\over2} {{2n+2} \choose {n+1}} \]
2
votes
1answer
263 views

Need help with Factorial Sums! [duplicate]

Possible Duplicate: How to prove that the number $1!+2!+3!+\dots+n!$ is never square? Show that the sum $$\sum_{k=1}^nk!\neq m^2$$for any integer $m$, for $n\geq4$.
3
votes
6answers
289 views

Show that the sequence $\left(\frac{2^n}{n!}\right)$ has a limit.

Show that the sequence $\left(\frac{2^n}{n!}\right)$ has a limit. I initially inferred that the question required me to use the definition of the limit of a sequence because a sequence is convergent ...
0
votes
1answer
106 views

Colored Blocks Factorial

I was given the following problem: A bag contains 14 red blocks, 10 white blocks and 12 blue blocks. How many different 19 block sets can be created from the collection in the bag? Two block sets ...
2
votes
1answer
137 views

Is this induction procedure correct? ($2^n<n!$)

I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form ...
4
votes
1answer
97 views

What's the arity of the factorial and exponential operations?

I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the ...
1
vote
2answers
239 views

First decimal digits of factorial $n$ divided by $x$

Actually, I will reformulate the question: how can I find a formula to calculate the nth decimal digit (the non-integer part) of f(x,n) = n!/x ? My idea is a Taylor serie of some kind but I don't know ...
4
votes
6answers
3k views

How can I calculate the limit of exponential divided by factorial? [duplicate]

I suspect this limit is 0, but how can I prove it? $$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$
4
votes
6answers
253 views

What is $n!$ when $n=0$? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles Why does 0! = 1? If I'm right, factorial $!$ means: $$n!=1 \cdot 2 \cdot 3 \cdot 4 \cdots n $$ so: $$ \begin{align} ...
2
votes
3answers
281 views

When is a factorial of a number equal to its triangular number?

Consider the set of all natural numbers $n$ for which the following proposition is true. $$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$ Here's an example: $$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
1
vote
3answers
69 views

Factorial number

Can anybody explain how $$ (k+2)(k+1)!-1 = (k+2)!-1 $$ also how $$ (k+1)!-1+(k+1)(k+1)! = [1+(k+1)](k+1)!-1 $$ my book show this example but i can't understand how. I also try google it but cannot ...