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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4
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3answers
84 views

Reducing $\prod \limits_{0 \le j \ne i \le n} \frac{n+1-j}{i-j}$ to $\frac{(n+1)!}{(n+1-i)\cdot i! \cdot (n-i)!}(-1)^{(n-i)}$

How could we show that: $$\prod_{0 \le j \ne i \le n} \frac{n+1-j}{i-j} = \frac{(n+1)!}{(n+1-i)\cdot i! \cdot (n-i)!}(-1)^{(n-i)} .$$ The module suggest we could reduce it by simply writing ...
4
votes
2answers
1k views

Factorial Moment of the Geometric Distribution

I am trying to caclulate the Factorial Moment of the Geometric Distribution #2 with parameter $p$. Therefore I set $\Omega = \mathbb{N}_0$ and have by using the pochhammer symbol and setting $q=1-q$ ...
3
votes
1answer
409 views

What is the theoretical upper bound of factorion numbers?

Recently I read about factorion numbers. I understood that there are only 4 factorion numbers, but what is the theoretical range in which they can be? Is it $[0, +\infty]$ or a smaller upper range? ...
32
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
11
votes
2answers
417 views

Given $n! = c$, how to find $n$?

I'm dealing with a time-complexity problem in which I know the running time of an algorithm: $$t = 1000 \mathrm{ms} .$$ I also know that the algorithm is upper bounded by $O(n!)$. I want to know ...
0
votes
4answers
285 views

Factorial of 0 - a convenience?

If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the ...
1
vote
5answers
2k views

Why does 0! = 1? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles Why does 0! = 1? All I know of factorial is that x! is equal to the product of all the numbers that come before it. The product of ...
22
votes
1answer
741 views

Repeated Factorials and Repeated Square Rooting

I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
2
votes
1answer
361 views

Use of algebra and factorials for a question related to proof by induction

$$ \begin{align*} &= (n+1)! − 1 + ( (n+1) · (n+1)! )\\ &= (n+1)! (1+n+1) − 1\\ &= (n+1)! (n+2) − 1\\ &= (n+2)! − 1\\ \end{align*} $$ I'm confused at how the first ...
2
votes
2answers
403 views

Combination vs Permutation?

This idea resulted while I heard an advertisement for Sonic, where they claim to have something like 300,000 different drinks they serve. Essentially, what they are allowing you do to is mix any soda ...
16
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5answers
1k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
2
votes
3answers
433 views

Arrangement of six triangles in a hexagon

You have six triangles. Two are red, two are blue, and two are green. How many truly different hexagons can you make by combining these triangles? I have two possible approachtes to solving this ...
11
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14answers
2k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
8
votes
4answers
832 views

How many consecutive composite integers follow k!+1?

I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's ...
0
votes
1answer
658 views

Logarithm base 2 and factorials

I'm learning about $\log_2$ for an algorithms class and theres a problem in the book that is confusing me. It asks: Find a formula for $\log_2(n!)$ using Stirling's approximation for $n!$, for large ...
7
votes
1answer
435 views

Is this a new formula?

It's late at night and I'm tired, but I just stumbled across this while doing my homework. Any chance this is new? Or, maybe, did I just somehow transform it and it is still basically the same ...
2
votes
1answer
184 views

How do I describe the growth of something that scales by a factorial?

I just wrote a blog post and wasn't sure how to word a particular sentence. Say I have the following function: \begin{equation} f(x) = x^2 \end{equation} Then I can say that the value of f(x) grows ...
5
votes
3answers
5k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
9
votes
2answers
306 views

Number of zeros not possible in $n!$ [duplicate]

Possible Duplicate: How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? The number of zeros which are not possible at the end of the $n!$ is: ...
40
votes
3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
0
votes
1answer
158 views

Show that $\Pi_{i < j} (v_i - v_j) \le k^{n^2}$ for $1 \le v_1 < v_2 < … < v_n = k$

Everything is in $\mathbb{Z}$. Let $v_1 < v_2 < ... < v_n = k$, and $v_1 = 1$ for $k >> n$. Let $ P = \Pi_{i < j} (v_j - v_i)$. How can I show that $P \le k^{n^2}$? There are $n + ...
3
votes
1answer
1k views

Factorial of a non-integer number

My TI-83 calculator doesnt allow me to do this, but using Windows calculator, I can compute the factorial of say 5.8. What does this mean and how does it work?
4
votes
2answers
957 views

How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?

I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these. First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = ...