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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7
votes
2answers
216 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
39 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
85 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
261 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
47 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
3answers
60 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
78 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
115 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
58 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
55 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
55 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
91 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 $n$ ...
0
votes
2answers
105 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
69 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
93 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
713 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
35 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?
12
votes
4answers
220 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
27 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
139 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
39 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
31 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
464 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
36 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
64 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 ...
0
votes
1answer
32 views

Constant multiplied by a factorial [closed]

I am doing an introduction to proof by induction and struggling to understand how this works: The highlighted bit is the area I am confused on. How can $(k+1)!+(k+1)[(k+1)!]−1$ become ...
1
vote
1answer
42 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
32 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
281 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
40 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
103 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
136 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
108 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
101 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
38 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) = ...
2
votes
1answer
53 views

Hat check problem. Ten friends total, five with sombreros, five with fedoras.

A group of ten people give their hats to the coatroom attendant. Five of the ten are wearing sombreros, and five and wearing fedoras. How many ways can the clerk return the hats so that no one gets ...
5
votes
3answers
55 views

Is there an efficient way to calculate $a! \bmod b$ for arbitrarily large $a < b$?

I am trying to find a way to calculate $a! \bmod b$ where $0<a<b$ for arbitrarily large $a$. Is there an efficient way to do this without calculating $a!$ explicitly?
0
votes
1answer
43 views

Is there a formula to find the exact value of inverse factorials?

$x!=y, \space x ∈ ℝ$ Is there a formula to find the exact value of $x$ in this case, assuming that we know the value of $y$? I could do $L(x)/W(\frac{L(x)}{e}) + \frac{1}{2}$ where $W$ is the ...
0
votes
1answer
26 views

How many ways to arrange 10 values in a vector of length 50?

I have a set of values $1,2,3,4,5,6,7,8,9,10$ that I want to place in vectors of length 50 in all permutations. So far I'm using the permutation formula with $n=50$, $r=10$ as $$ \frac{50!}{(50-10)!} ...
1
vote
4answers
62 views

In layman's terms, what is a multifactorial? - $x\underbrace{!!\cdots!}_{n\text{ times} }$

My first impression was that given a multifactorial expression, one was to factorial the first term, then factorial that term, then factorial that term, etc. etc. So, 20!! = (20!)! I now understand ...
3
votes
3answers
80 views

How to find $\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}$

What method should I use for this limit? $$ \lim_{n\to \infty}{\frac{n^{n-1}}{n!}} $$ I tried ratio test but I ended with the ugly answer $$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}} $$ which ...
1
vote
0answers
27 views

Explaination of how Wallis's formula was used to arrive at the end result?

While looking for a proof on Stirling's formula I am having a little trouble connecting the dots between these last two steps here: [1]: ! The first formula is Wallis's formula.
1
vote
3answers
80 views

Limit as $n$ goes to infinity of $n^22^n/n!$ [closed]

I have read in a book that $( n^2 2^n)$ is superior to n! this means that the limit below will at least be a constant and $n! = O( n^2 2^n )$, but l could not manage to find it, any ideas ...
2
votes
2answers
33 views

Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
-3
votes
3answers
56 views

How to simplify infinite multiple? [closed]

I have a series like $$1!2!3! ... n!$$ for some finite n. How do I write this in notation? Similar to summation I guess or if it can simply be reduced down to something in terms of n that would be ...
4
votes
2answers
61 views

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $

What is the value of $n$ for which $n!=2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23 $ The way I am approaching this problem is just to find the ...