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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22
votes
4answers
894 views

Solutions for x!/y!=(y+1)!

I was watching a video recently, and I saw how 10*9*8*7 was equal to 7*6*5*4*3*2*1, or to make it clearer, 10!/6!=7!. I was wondering if there were any other solutions, so I checked the web, to find ...
0
votes
3answers
39 views

Simlifying [(k+1)! - 1] + (k+1)((k + 1)!)

I'm afraid I've gotten a bit rusty on Math since I was last in university. I was looking at a problem in my text book that simplified ...
0
votes
1answer
32 views

Simplify the following problem

How $$\frac{1}{k}\sum_{m=r}^{k-1}\frac{m!}{(m-r)!}=\frac{(k-1)(k-2)\ldots(k-r)}{r+1};\quad r=1,2,\ldots$$ I have thought in two ways: ...
0
votes
1answer
20 views

Simple Calculation

How $$\sum_{m=r}^{k-1}m(m-1)\ldots(m-r+1)=\sum_{m=r}^{k-1}\frac{m!}{(m-r)!};\quad r=1,2,\ldots$$ ? i have ...
3
votes
1answer
96 views

The number of zeros in the decimal representation of the factorial of 126

How many zeros are in $126!$ ... the result is $34$. But can I calculate it manually? I have seen How many zeroes are in 100! but I don't think it's helpful.
1
vote
3answers
80 views

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$.

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$. I get $$\frac{1}{r!}-\frac{1}{(r+1)!}=\frac{(r+1)-r!}{r!(r+1)!}$$ and in the numerator since $$(r+1)!-r!=r$$ so ...
2
votes
2answers
882 views

How many ways to split 5 numbers in 2 groups?

How many ways can you split the numbers 1 to 5 into two groups of varying size? For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is ...
1
vote
0answers
41 views

What are the best and most elementary bounds for $n!$?

What this question is looking for is bounds on $n!$ that are elementary in nature (I seem to have a fetish for these type of proofs). In general, as the results become more complicated, they also ...
6
votes
1answer
365 views

Solving equation involving factorial

I have the following problem: $$ N^n(N-n)!=A $$ Where $N$ and $A$ are constants. I want to solve this equation for $n$ (for a variation of the birthday problem), but I have little experience with ...
2
votes
2answers
133 views

Is it easier to calculate a factorial or the inverse of a factorial (1/n!) for extremely large n?

I need to calculate extremely large factorials but they grow extremely fast! I was wondering if it might be easier to calculate $\frac{1}{n!}$ rather than n! itself because as n goes to infinity ...
-1
votes
3answers
108 views

Prove that $n! \geq 2^{n-1}$ for $ n\geq1$ [duplicate]

Mathematical Induction:-Prove that $n! \geq 2^{(n-1)}$ for $n\geq 1$. I tried mathematical induction but could not
1
vote
4answers
123 views

How many values of $n<50$ exist that satisfy$ (n-1)! \ne kn$ where n,k are natural numbers?

How many natural numbers less than 50 exist that satisfy $ (n-1)! \ne kn$ where n,k are natural numbers and $n \lt 50$ ? when n=1 $0!=1*1$ when n=2 $1!\ne2*1$ ... ... ... when n=49 ...
0
votes
2answers
104 views

Can 44 be a factor of $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)$ where n is a natural number greater than 3?

Can 44 be a factor of $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)$ where n is a natural number greater than 3 ? When $n =4 $, $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7$
2
votes
0answers
96 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
1
vote
1answer
336 views

How to calculate the number of permutations and combinations if k is equal to n?

Say the question is How many unique ways are there to arrange the letters in the word FANCY? The formula I use for permutations is n! / (n - k)! ...
4
votes
2answers
101 views

Approximation of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $

I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right! ...
2
votes
1answer
63 views

Limit of a function with factorial [duplicate]

$\lim_{n \rightarrow \infty} (x^n/n!)=0$. prove. x is finite whereas n is infinite. But increasing n means also increasing $x^n$. It is understandable that if n is too large n! will exceed $x^n$. How ...
3
votes
2answers
439 views

German tank problem, simple derivation [duplicate]

I was reading the recent question on the German tank problem, and had trouble with one of the derivations in this section. $$\sum_{m=k}^N m \frac{\binom{m-1}{k-1}}{\binom N k} = ...
5
votes
3answers
201 views

How Many Ways to Build a 6-Pack

There is a beverage company here that claims to have a selection of 200 different beers. They have a special deal where you can build your own six pack at a discount. They advertise that there are ...
1
vote
1answer
69 views

How many digits are 9 at the end?

How many digits at the rightmost of this sum are 9? $$1! + 2\times2! + 3\times3! +\dots +48\times48!$$ I tried to calculate the first few terms but I couldn't solve it. The answer is 10.
1
vote
1answer
2k views

How do we calculate factorials for numbers with decimal places? [duplicate]

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\times1$, but how do we ...
1
vote
0answers
65 views

Limit of the sequence $\frac{(n!)^{1/n}}{n}$ [duplicate]

Which is the limit of the fllowing sequence $$\frac{(n!)^{1/n}}{n}$$
2
votes
2answers
227 views

Limit, factorials

There is the following limit, I would like to calculate: $\lim_{n\rightarrow\infty}\frac{n!}{\left(n+1/6\right)!}$ I tried to use the Stirling approaximation formula $n!\approx\sqrt{2\pi ...
6
votes
4answers
490 views

Find the sum of series $\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}$

I wanted to know how can I start to find the sum of the series: $$\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}=\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}\cdots$$ I am having no clue. Thanks.
14
votes
11answers
2k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
1
vote
3answers
123 views

Product representations of the factorial function?

Is this the only product representation of the factorial function? $$ {n!} =\prod_{k=1}^{n} k $$
9
votes
3answers
4k 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 ...
0
votes
1answer
43 views

Non-equality with Gamma functions

Let $n \in N$, $k \in Z_+$. Show that $$ \frac{\Gamma\left(k+\frac 12\right)}{\Gamma\left(k+\frac ...
2
votes
2answers
661 views

how to find remainder when $20! + 20^{23}$ is divided by $23$?

how to find remainder when $20! + 20^{23}$ is divided by $23$? I am finding it bit difficult to solve. Does any one has a simpler way to solve this problem??
-2
votes
1answer
185 views

Sigma of factorial function? [duplicate]

How would you find this sum mathmatically?$$\sum_{i=0}^\infty \left( \dfrac{1}{2} \right) ^{i!}$$ What techniques would you use to solve this as well?
0
votes
2answers
236 views

Factorial related problems

How many zeros are there in $ 25!$ ? My answer was $6$. But i solved it by finding how many numbers are divisible by $5$ and $2$.here i was told to find out the zeros at the last end. But what is the ...
1
vote
2answers
248 views

Integrating Factorials

I feel like I'm doing something wrong here: $$\frac{d^n}{dx^n}(x^n)=n!$$ $$ 5!=\frac{d^5}{dx^5}(x^5)$$ $$ \int{5! dx}=\int{\frac{d^5}{dx^5}(x^5)}dx=x\frac{d^4}{dx^4}(x^4)=x*4!$$ Please explain what ...
0
votes
2answers
47 views

Simplyfing Probability equation

I was solving a homework problem, and I had obtained a formula for the required probability in the question. What I wanted to ask could it be more simplified? $$P = \sum_{i=0}^{a}( \frac{a!}{(a-i)!} * ...
6
votes
3answers
149 views

Proving a complex sum equals factorial

I have just stumbled across the equality that: $$ \sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n} \binom{n}{j} = n! $$ How would I go about proving this equality? Also, what is the left hand side equal to if ...
1
vote
1answer
77 views

I can not figure out the answer to this greatest integer problem

I can’t figure out the answer to this problem: $$\left\lfloor\sqrt{\lfloor\pi\rfloor!}\right\rfloor=\;?$$
7
votes
3answers
326 views

Is there an inverse to Stirling's approximation?

The factorial function cannot have an inverse, $0!$ and $1!$ having the same value. However, Stirling's approximation of the factorial $x! \sim x^xe^{-x}\sqrt{2\pi x}$ does not have this problem, and ...
1
vote
1answer
51 views

At what $n$ does $\sum_{k=0}^\infty{(n\ k)!\over (k!)^3}$ diverge?

Consider $$\sum_{k=0}^\infty{(n\ k)!\over (k!)^3}$$ where $0<n$. For what $x$ where $0<x\leq n$ does the sum diverge?
9
votes
2answers
177 views

Polynomials mapping factorials to factorials

I'm looking for all polynomials $P(x)$ with integer coefficients such that for every $n \in \Bbb N$ there is an $m \in \Bbb N$ such that $P(n!)=m$!. The only solutions seem to be the constant ...
1
vote
0answers
66 views

When is $n!+1$ a square? [duplicate]

I'm looking for the solutions $(n,m)$ of the equation $n!+1=m^2$. I have calculated the values of $\sqrt{n!+1}$ for $n \le $ and found only the solutions $(4,5)$, $(5,11)$ and $(7,71)$. Are these ...
12
votes
1answer
323 views

Yet another $\sum = \pi$. Need to prove.

How could one prove that $$\sum_{k=0}^\infty \frac{2^{1-k} (3-25 k)(2 k)!\,k!}{(3 k)!} = -\pi$$ I've seen similar series, but none like this one... It seems irreducible in current form, and I have ...
2
votes
1answer
68 views

Why is this formula for $(2m-1)!!$ correct?

Numerically calculating the sum of the squares of the $m$th row of Pascal's triangle, I found that for at least the first $10$ or so cases $$\sum_{i=0}^m \binom{m}{i}^2=\frac{(4m-2)!!!!}{m!}$$ Where ...
1
vote
0answers
65 views

Simplify $\frac{[m+n-1]!}{[m]![n]!}$ where $[k]=x^k-x^{-k}$ and $[k]!=[2][3]…[k]$.

Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
4
votes
4answers
247 views

$\left(-\frac{1}{2}\right)! = \sqrt{\pi}?$ [duplicate]

I recently learned that $\left(-\frac{1}{2}\right)! = \sqrt{\pi}$ but I don't understand how that makes sense. Can someone please explain how this is possible? Thanks!
0
votes
3answers
116 views

Proof regarding factorials.

Suppose $a$ and $k$ are positive integers, then how would you prove(not intuitively) that: $a!k! \leq (ak)!$ Although it is apparent that the inequality is correct, but how can I show this ...
1
vote
1answer
89 views

Prove that $ \lim_{n \to \infty}\frac{n}{\sqrt [n]n!}=e$? [duplicate]

I am stuck on the following problem : Prove that $\lim \limits_{n \to \infty}\frac{n}{\sqrt [n]n!}=e$ ? Can someone point me in the right direction? EDIT: It was actually a part of a ...
1
vote
1answer
123 views

Help with a different approach to extracting a polynomial equation from differences

It is well known that we can determine the degree of a polynomial can be found by finding when the differences are the same. i.e. if the second differences are the same, it is a polynomial of the 2nd ...
1
vote
2answers
30 views

Does $n=n^2 - (n!\;\bmod n^2)\implies\text{isPrime}(n) = \text{True}$?

With integers $n$, of such form that $$n=n^2 - (n!\mod n^2)$$ Is $n$ always a prime number?
5
votes
3answers
266 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
3
votes
3answers
345 views

Evaluate $\displaystyle\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
1
vote
0answers
83 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...