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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4answers
130 views

What is the meaning of $(2n)!$

I came across something that confused me $$(2n)!=?$$ What does this mean: $$2!n!, \quad 2(n!)$$ or $$(2n)!=(2n)(2n-1)(2n-2)...n...(n-1)(n-2)...1$$ Which one is right? The exercise is to show that ...
0
votes
1answer
14 views

Series increasing or decreasing with factorials

I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are ...
0
votes
3answers
71 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
0
votes
1answer
40 views

Computing $\mathrm{gcd} (100!, 3^{100})$

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem ...
1
vote
3answers
85 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
1
vote
1answer
69 views

Proving an identity involving factorials

I have stumbled upon the following statement and have verified it computationally for many $n$ (up to n=500, it took a long time for my computer to do out all of the math), yet I have no idea how to ...
3
votes
1answer
35 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
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 $100!$ ($5$ ...
1
vote
2answers
32 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
3
votes
2answers
876 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
0
votes
3answers
46 views

Simplifying factorials

I apologise for a repost, but my rep is not high enough to ask in a comment. But, in this question Simplify sum of factorials with mathematical induction I am confused how: $$(n+1)!-1+(n+1)(n+1)! = ...
3
votes
2answers
1k views

Proof the inequality $n! \geq 2^n$ by induction

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the ...
0
votes
1answer
38 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
0
votes
1answer
102 views

Finding the largest factorial of only three digits

I am using the following Python code to compute the above, but no results up to 16000!: ...
9
votes
5answers
6k views

Find the sum of the digits in the number 100!

I am working on a Project Euler problem http://projecteuler.net/problem=20. $n!$ means $n(n - 1)\dots...3.2. 1.$ For example, $10!$ $=$ $10$ $9$ $...$ $3$ $2$ $1$ $=$ $3628800$, and the ...
1
vote
2answers
56 views

$x!=y^n$ for $x,y \neq 0,1$

A straightforward problem (find all integers such that $m!+3=n^2$) led me into thinking about the integers for which: $$x!=y^2$$ is true. I argued that other than the trivial case ($x!=1$) that this ...
0
votes
2answers
130 views

Inequality $(n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n$

Prove that $$ (n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n $$ holds for all $n\in\mathbb{Z^+}$. I tried induction but there's no obvious way to go from $n$ to $n+1$.
3
votes
3answers
71 views

Proving $\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$ is an integer

I have verified this for many values of $n$, but I have no idea how to prove it. Does anyone know how I could go about showing that: $$\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$$ is an integer when ...
0
votes
0answers
16 views

Function to define how combinations N items can be organized with a certain condition

This is not a factorial only problem If I have 5 items and I wanted to know how many possible ways they could be arranged, the answer is 5! or 120. However my situation is I need to know how many ...
11
votes
2answers
691 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
2
votes
1answer
20 views

Is there a way to express $(n-i)!(n-j)!(2i)!(2j)!$ in terms of $n$ and $r=i+j$?

I have been attempting to simplify the double sum: $$\sum_{i=0}^n \sum_{j=0}^n \frac{(-1)^{i+j} (2i+2j)!}{(n-i)!(n-j)!(2i)!(2j)!2^{i+j}(i+j)!}$$ And so what I am attempting to do is rewrite it in ...
1
vote
2answers
384 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
3
votes
2answers
105 views

Summing of factorials to produce perfect cubes

I was playing around with factorials the other day, and I realized that $4!+5!$ is a perfect square. Perplexed by this result, I started looking for other pairs of factorials that produce a perfect ...
1
vote
0answers
42 views

Limit of $\frac{1}{n}\log({n\choose np})$ without using Stirling's formula

I am trying to evaluate the following limit: $$ \forall p\in(0,1) ,\lim_{n\rightarrow \infty} \frac{1}{n}\log{n\choose \lfloor np \rfloor} =H(p),$$ where $\lfloor x\rfloor$ means the greatest integer ...
4
votes
3answers
56 views

Proving $ n! \geq 2^{n-1} $

Prove that $$ n! \geq 2^{n-1}$$ for $n \geq 1$. My initial solution by induction goes like this. For $n = 1 : 1 \geq 1 $. Assuming that $$ n ! \geq 2^{n-1}.$$ Then for $n+1$, $$ (n+1)! = ...
11
votes
1answer
158 views

Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2

Prove that: If $n$ is greater than 2, then $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ From Barnard & Child's "Higher Algebra". I know that the highest power of a prime $p$ ...
2
votes
3answers
53 views

Evaluate: $\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$

Evaluate: $$\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$$ Attempt $S_2=\frac {n!}{(n-2)!}$ $S_3=\frac {n!}{(n-3)!}$ $S_4=\frac {n!}{2(n-4)!}$ $\vdots$ $S_{n-1}=\frac {n!}{1!(n-3)!}$ $S_n=\frac ...
4
votes
2answers
101 views

Show that $p!$ and $(p - 1)! - 1$ are relatively prime

If $p$ is prime number, with $p>3$ Show that $p!$ and $(p - 1)! - 1$ are relatively prime. I tried $\text{gcd}\;(p!,(p-1)!-1)=d\Longrightarrow d\mid p!$ e $d\mid(p-1)!-1$ having ...
36
votes
3answers
3k views

Is $n! + 1$ often a prime?

Related to this question, I wonder how often $n!+1$ is a prime? There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not... or anything in that sense...
1
vote
0answers
136 views

Sum concerning Catalan triangle and binomials

I am trying to prove the following relation. For $k \in \mathbb{N},k \geq 2$ and $i=1,\ldots,k-1$ \begin{equation} {2k-2-i \choose k-1-i}=\sum \limits_{l=0}^{k-1-i} 2^{2l} T(k-2-l,k-1-i-l)-2 \sum ...
3
votes
4answers
58 views

Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

Is there a proof for the following identity that only uses the definition of the (generalized) binomial coefficient and basic transformations? Let $n$ be a non-negative integer. $$\binom{2n}{n} = ...
1
vote
1answer
30 views

Is there a closed form for this sequence?

I'm trying to find a closed form for the following sequence: $a$ $a(a-1)$ $a(a-1)(a-2)$ $a(a-1)(a-2)(a-3)$ The problem is, $a=\frac{1}{2}$. If it were some whole number, then I'd use ...
8
votes
5answers
341 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
1
vote
2answers
72 views

Question about Binomial Sums [duplicate]

Prove that for any $a \in \mathbb{R}$ $$\sum_{k=0}^n (-1)^{k}\binom{n}{k}(a-k)^{n}=n!$$ I rewrote the sum as $$\sum_{k=0}^n \left((-1)^{k}\binom{n}{k} \sum_{i=0}^n (-1)^{i}a^{n-i} k^{i} ...
18
votes
5answers
4k views

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce ...
1
vote
1answer
125 views

Need help finding a closed form for complicated sum

I'm trying to find a closed form expression for the following sequence: $$a_n=\sum_{i=1}^{n}\frac{(n-1-i+d)!}{(n-2i)!(i)!}=\sum_{i=1}^{\frac{n}{2}}\frac{(n-1-i+d)!}{(n-2i)!(i)!}$$ Where $n$ and $d$ ...
4
votes
2answers
64 views

How to calculate what this power series converges against? (double factorials)

I'm working on my physics master course homework and I'm given the following equation out of nowhere: $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over ...
-2
votes
1answer
400 views

Even or Odd for factorial

Moderator Note: This was a contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this question? ...
3
votes
0answers
116 views

Sum problem involving Factorials

got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$: \begin{equation} \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! ...
1
vote
0answers
19 views

What condition does a recursive function have to fulfill to be well defined?

What condition does a recursive function have to fulfill to be well defined? Provide a well-defined recursive definition of the factorial of a number. Modify the definition so that is no longer ...
3
votes
1answer
24 views

How to simplify a fraction involving factorials

I have following term: $$\frac{\frac{3^{2k+2}}{(2k+2)!}}{\frac{2^{2k}}{(2k)!}}=\frac{3^{2k+2}\cdot(2k)!}{(2k+2)! \cdot 3^{2k}}=9\cdot\frac{(2k)!}{(2k+2)!}$$ I know that you can simplify even further ...
15
votes
15answers
3k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
1
vote
3answers
76 views

convergence of $\sum_{n=2}^{\infty}\frac{(n+1)!(n+1)^{n-1}}{n^{2n}}$

$$\sum_{n=2}^{\infty}\frac{(n+1)!(n+1)^{n-1}}{n^{2n}}$$ I used the Cauchy test and it lead me to $\frac{\sqrt[n]{n!}}{n^2}$. But I can't tell what is the limit of this. I tried the Squeeze theorem: ...
3
votes
4answers
100 views

Solving for $r$ in ${12\choose{r}}=924$

I can solve the equation $_{12}C_r=924$ fairly easily by guess and test because there are so few possible $r$ values, but is there a clean way to solve an equation of this format algebraically? I ...
2
votes
1answer
52 views

Limit of $\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!}$

For a given $n \geq 1$ $\hspace{2mm}(n \in \mathbb{R})$, I know that $$\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!},$$ only exists and it is equal to zero if $a \geq n.$ However, I cannot ...
7
votes
1answer
4k views

When does a Square Matrix have an LU Decomposition?

When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
21
votes
2answers
15k views

How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a ...
1
vote
0answers
24 views

Permutations and combinations - divisibility of a factorial

Question: Find the largest value of $n$ for which $125!$ is divisible by $6^n$ Approach: I tried to find all the numbers which were a multiple of 6. The number of times such divisors occurred ...
1
vote
2answers
41 views

Number of zeroes at end of factorial

Question: How many zeroes will there be at the end of $(127)!$ Approach: Considering the fact that when two numbers ending in $x$ and $y$ zeroes are multiplied, the resulting number contains $x+y$ ...
35
votes
14answers
4k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...