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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8
votes
3answers
114 views

Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
0
votes
2answers
554 views

Powerball odds - factorial?

According to Powerball.com, the game is played like this ...we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls Their odds explain that the ...
2
votes
8answers
136 views

Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

The following series $$\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$$ converges. It fails the divergence test, but once I apply the ratio test, the limit is always equal to $\infty$. Unless you cannot ...
0
votes
1answer
44 views

identity with falling factorials

How can one show that $$\sum_{k=0}^n \frac{(n)_k}{k!} = 2^n$$ for all $n \geq 0$ where for $m \in \mathbb{Z}$ and $k \geq 0$ $(m)_k$ is the "falling factorial": $$(m)_k = \begin{cases} 1, ...
0
votes
1answer
55 views

Fraction Factorial [duplicate]

How do we find factorial of fractions? For eg: $\frac{1!}{2!}=(\frac{\pi}{4})^{\frac{1}{2}}$ Factorials are used in combinatorics and they can only be functioned on integers to give integers.Then how ...
0
votes
1answer
39 views

highest value of 'a'

I got a question when I started factorials Q. If $a^8$ and $8^a$ is completely divisible by $50!$ Then which one of the following is true about 'highest value of a'? (A) $10<a<14$ ...
2
votes
2answers
49 views

Factorial Representation of product

So I've been trying to work out if it is possible to write: $\large \Pi_{i=1}^n (3i-1)$ as an expression involving the quotient or product of two factorials, or really any expression involving ...
4
votes
0answers
45 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
2
votes
0answers
40 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
6
votes
4answers
114 views

Prove: $\forall$ $n\in \mathbb N, (2^n)!$ is divisible by $2^{(2^n)-1}$ and is not divisible by $2^{2^n}$

I assume induction must be used, but I'm having trouble thinking on how to use it when dealing with divisibility when there's no clear, useful way of factorizing the numbers.
0
votes
0answers
57 views

How many trailing zeroes does 4617! contain? [duplicate]

I am getting $1151$ as answer on continuous division by $5$. Is it right? On each division by 5, some remainder is generated...doesn't that count? Example: 4617/5 + 923/5 + 184/5 + 36/5 + 7/5 ...
0
votes
1answer
54 views

Can a factorial have an exponent

can you do $(4!)^4$ or do you have to do $4! \times 4$, I have looked on google but can't find anything to prove or disprove if it's possible.
7
votes
2answers
226 views

Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer

I need help to prove that the quantity of this division : $\dfrac{({10!})!}{{10!}^{9!}}$ is an integer number, using combinatorial proof
1
vote
0answers
29 views

Permutations to divide a solid

I have a 3-dimensional cuboid, with dimensions 2x2x1. I wish to divide this into smaller EQUAL sized cuboids of size 2x1x1. Then I want to extend this case for larger cuboids, assuming that equal ...
0
votes
1answer
40 views

Is there a way to show X equal or smaller than Y?

X=$1!+2!+3!+...+n!$ Y=$n+n(n-1)+n(n-1)(n-2)+....+n(n-1)(n-2)(n-3)...(n-a)$ Is there a way to show X equal or smaller than Y? for $a= n-1$ and and $a=n-2$ i found that ...
-1
votes
2answers
102 views

Finding last two non-zero digits of 2016! [closed]

Find last two non-zero digits of 2016! I wasn't able to find anything which could help me. I need the method to solve it in the "mathematical" way, without computing the factorial itself.
-1
votes
1answer
56 views

Derivative of the function $(x)!$. [duplicate]

I had been learning calculus. So what I was thinking about is what us differentiation if $(x)!$. I know. 'n 'th derivative of $x^n$ is $x!$ but it isn''t helping me to solve this problem.
-1
votes
3answers
55 views

Prove: $4^{2k+1}>1 \cdot 3 \cdot \dots \cdot (2k+1)$ where $k$ is a positive integer

Prove: $$4^{2k+1}>1 \cdot 3 \cdot \dots \cdot (2k+1)$$ where $k$ is a positive integer. The difference is very large but I cannot find any way to prove it.
2
votes
2answers
294 views

Verify If Sum of Factorials is Divisible by Integer

I am working on preparing for JEE and was working on this math problem. We have the sum, $$\sum_{n=1}^{120}n!=1!+2!+3!+\ldots+120!$$ Now I am given the question, which says that what happens when ...
2
votes
3answers
54 views

How to decide if a factorial is a multiple of certain number? [closed]

How to decide if a factorial is a multiple of certain number? For example, if I have to decide whether $123!$ is a multiple of $4$ or not what should be the procedure?
2
votes
4answers
84 views

How to compute $\lim\limits_{x \to +\infty} \left(\frac{\left((x-1)^2-x\ln(x)\right)(x)!}{(x+2)!+7^x}\right)$?

I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Can you explain the method and the steps used? $$\lim\limits_{x \to +\infty} ...
4
votes
1answer
83 views

Find $\lim\limits_{n \to \infty} \frac{1!+3!+\ldots+(2n-1)!}{2!+4!+\ldots+(2n)!}$

$$\lim\limits_{n \to \infty} \frac{1!+3!+\ldots+(2n-1)!}{2!+4!+\ldots+(2n)!}$$ I have tried some standard approaches like dividing by $(2n)!$ and comparing consecutive terms. Hint, please :) ...
1
vote
2answers
116 views

How to find 10th digit of $\sum_{k=1}^{49} k!$

How to find the tenth digit (from the right) of $\sum_{k=1}^{49} (k!)$ analytically. The worst possible method would be to actually sum each individual number which would yield a number of order ...
1
vote
2answers
64 views

Inequality proof involving multinomial coefficients

How may I proceed to prove/disprove following inequality? $$\frac{n^n}{p_1^{p_1}\cdot p_2^{p_2}\cdots p_k^{p_k}}>\frac{n!}{p_1! p_2!\cdots p_k!} $$ where $\sum_{i=1}^k p_i=n$ It seems, using ...
0
votes
1answer
56 views

Number of solution of $x\cdot (p-1)! \equiv x\pmod {np}$ [closed]

Find number family of solutions of congruence $$x\cdot (p-1)! \equiv x\pmod {np}$$ Where, $p$ is a prime number.
1
vote
2answers
57 views

Let $f(x) = x^n$. Show that $f^{(n)} = n!$ and $f^{(m)} (x) = 0$ for all $m > n$.

I'm supposed to use mathematical induction to solve this Show that $P(1)$ is true Assume $P(K)$ is true Show that $P(K+1)$ is true How do I approach this problem?
0
votes
1answer
105 views

Finding maximum number of factors in n!

I am quite new to $v_p()$ problems, and would like to know if anyone prove that $v_n(n!)\le n/2$? Basically, what I mean is that prove that for all positive integers $n$, the amount of factors of $n$ ...
1
vote
2answers
123 views

Formula for $\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!}$

Is there a closed form for the infinite sum $$\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!} \mathrm{?}$$ where a is an integer greater than or equal to $0$. When $a=0$, the sum is just the series ...
4
votes
1answer
72 views

Double Summation Over all subset of $\{1,2,…n\}$

In Benson's Book "Polynomial In variants of Finite Groups" It is claimed that(Without any proof): $$ j! u_1u_2...u_j =\sum_{I \subseteq \{1,2,...,j\} } (-1)^I (\sum_{i \in I}u_i)^j$$ Where $I$ runs ...
2
votes
1answer
95 views

Summation relating factorial and cosine

How to simplify \begin{align*} \sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{\left(2k\right)!}{4^{k}\left(k!\right)^{2}}\cos\left(kx\right) \end{align*} for $0\leq x <\pi$ ? I don't even know where ...
5
votes
5answers
762 views

Prime factors of a factorial [closed]

Determine all (distinct) prime factors of $1000!$. Here we seek a description of these factors as a set; there is no need to compute them. What exactly do I need to determine here?
0
votes
4answers
38 views

Dividing factorials

I'm told that $\dfrac{(n+1)!}{(n+2)!}$ simplifies to $\dfrac{1}{n+2}$, but I dont understand how this works. Could someone explain the theory of how to divide factorials like this?
25
votes
8answers
609 views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ ...
0
votes
1answer
29 views

How to prove that this upper bound for $n!$ will always work for suitable $n_0(k)$?

So, I know that $n!<(\frac{n}{2})^n$ for $n\geq6$. The natural question that arises is: Is it true that for every $k \in \mathbb N$ there exists $n_0(k)$ such that $n!<(\frac {n}{k})^n$ for ...
5
votes
2answers
162 views

Is it possible to compute factorials by converting to matrix multiplications?

An $n$-th term of the Fibonacci sequence can be computed by a nice trick by converting the recurrence relation in a matrix form. Then we compute $M^n$ in $O(\log n)$ steps using exponentiation by ...
2
votes
1answer
40 views

Alternative factorization of $\prod\limits^{n}_{k=1}k!^{k+1}$

Question: How can I succinctly express (using the product and sum notations) the following expression? $$n^{(n+1)}(n-1)^{(n+1)+n}(n-2)^{(n+1)+n+(n-1)}\cdot\cdot\cdot ...
1
vote
1answer
37 views

Is there a way to find the number of trailing zeroes in a factorial with a certain base?

I have a number $k$, and I need to find the number of trailing zeros $k!$ (factorial) has when put into base $b$. I need a general way that will work for all $b$'s and $k$'s. I have tried making ...
7
votes
3answers
483 views

After 6n roll of dice, what is the probability each face was rolled exactly n times?

This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible ...
0
votes
2answers
39 views

estimate of n factorial

on our lesson at our university, our professsor told that factorial has thie estimates $n^{\frac{n}{2}} \le n! \le \left(\dfrac{n+1}{2}\right)^{n}$ and during proof he did this ...
3
votes
2answers
77 views

What is the closed form approximation of the asymptotic growth rate of the superfactorial function?

The asymptotic growth rate of the hyperfactorial function (defined to be: $H(n)=\prod^n_{k=1}k^k$) is apparently (approximately) equal to: I'm curious as to how this result is obtained, and am ...
1
vote
1answer
51 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
0
votes
2answers
36 views

Solving Large Factorial Division without writing out factorials

I am calculating entropy for a physics problem and it requires solving this equation: $\ Entropy = \frac{949!}{899! 50!} $ However, I am not sure how to solve this mathematically without reverting ...
3
votes
2answers
290 views

Factorials in different base

Got an interesting problem from a friend. How many zeroes does $n!$ end in when written in base $n$? For every factor of $n$ in $n!$, I know that there will be $1$ $0$ added. However, I'm not really ...
1
vote
2answers
44 views

How do I simplify this equation

I'm trying to find a formula that will allow me to calculate the sum total of a progression (not sure if that's the word) in a spreadsheet. $$1 + 0.79 + 0.79\cdot 0.79 + 0.79\cdot 0.79\cdot 0.79 ...
2
votes
3answers
105 views

How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$

$$\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$$ Can someone show why this estimate holds true? I tried quite a bit but couldn't really find a way to approach this. WolframAlpha says it is true ...
3
votes
1answer
141 views

Uniquely identify any finite subset of an infinite set

Let $U$ be an unbounded subset of $\mathbb{N}$. Let $D = \mathcal{P}_{<\omega}(U)$ (the set of all finite subsets of $U$). Let $f$ be an injection such that: $f: D \rightarrow \mathbb{N} $ ...
0
votes
1answer
23 views

Factorial Series Written As Recursive Definition

I have a factorial series as shown below: \begin{equation} (2n+1)!~\text{for all $n \geq 0$} \end{equation} And I would like to know if the recursive definition that I wrote is accurate: ...
4
votes
0answers
110 views

Evaluate $\lim\limits_{n\to \infty} {\frac{(2n-1)!!}{(2n)!!}}$ [duplicate]

I have tried using Stolz–Cesàro's formula and subtract the next term in the series but that gave me $$\lim_{n\to \infty} {\frac{2n}{2n+1}}$$ which is obviously 1 and not right. I do realise that the ...
5
votes
4answers
114 views

Prove $n! \leq n^n$

Prove by least counter example for all positive integers n. $$ n! \leq n^n$$ I keep getting stuck after proving the least element of the set of counterexamples can not equal 1. Any suggestions would ...
0
votes
0answers
45 views

proof of multi choose equivalence

I could really use some help proving this. Let n and k be positive integers, and let $\left(\!\!{n\choose k}\!\!\right) ={n+k -1\choose k}$, prove: $$\left(\!\!{n\choose k}\!\!\right) = ...