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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1answer
35 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 ...
1
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2answers
79 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} ...
4
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2answers
101 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 ...
3
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0answers
148 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)! ...
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0answers
22 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 ...
1
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1answer
134 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 $d$ ...
3
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1answer
34 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 ...
1
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0answers
158 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
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4answers
112 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
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1answer
58 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 ...
1
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3answers
79 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: ...
1
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0answers
28 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 ...
2
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2answers
61 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 $x+y$ ...
2
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0answers
110 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
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3answers
202 views

For all $n>2$ there exists a prime number between $n$ and $ n!$

How to prove that there exists a prime number between $n$ and $ n!$, for all $ n> 2$? (Bertrand's postulate gives a much better bound, but this question is about obtaining a self-contained ...
1
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1answer
82 views

general formula for permutations with some ordering

Assume I want all permutations of a set of numbers with certain numbers must go before others. Similar to this question but I'm looking for a more general formula. For example the set ...
0
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1answer
43 views

Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

I'm trying to prove that $$\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}\left[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}\right]$$ for $p,n \geq 2$, $p, q \in \mathbb{N}$. I'm trying to use induction ...
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3answers
366 views

Other representations of factorial

I have a little question: which other representation of factorial $n!$ without using the factorial? Is there any definition of factorial as a series? or any other way? Thanks in advanced.
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4answers
246 views

How many digits are there in 100!? [closed]

How many digits are there in 100 factorial? How does one calculate the number of digits?
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3answers
85 views

Writing number as sum of reciprocals of factorial

Given a real number $r>0$. Is there a way to determine whether $r$ can be written as a (possibly infinite) sum of distinct terms of the form $1/n!$? For example, if we want to determine whether ...
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3answers
96 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, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
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1answer
333 views

Help with difficult telescoping series question

Evaluate $$\frac3{1!+2!+3!}+\frac4{2!+3!+4!}+\ldots+\frac{2012}{2010!+2011!+2012!}\;.$$ I see that the question is telescoping, but I don't know how to break it down into a form similar to that of ...
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1answer
28 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
1
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3answers
43 views

What is the growth rate of the logarithm of the factorial sequence?

I'd like to know the space complexity of storing bit string representations of the numbers in the factorial sequence (as in a memoized factorial function). So each number $f_i=i!$ in $i=0 \cdots n$ ...
3
votes
2answers
133 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 ...
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0answers
66 views

Are there infinite primes of the form n!+1? [duplicate]

Are there infinite primes of the form n!+1 ? I searched the internet but couldn't find the answer.
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3answers
35 views

The only solution of the equation ${72_8!}/{18_2!}=4^x$ is $x=9$

Problem and Definitions If $n_a!:=n(n-a)(n-2a)(n-3a)\ldots(n-ka):n>ka$, how should I go about solving this?: $$\dfrac{72_8!}{18_2!}=4^x$$ Attempt ...
3
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4answers
198 views

Formula for $1! \times 2! \times \cdots \times n!$?

Are there any useful forms for the expression $1!\cdot 2!\cdot 3!\cdot ...\cdot n!$? I'm trying to solve a problem that involves this expression and thought it might help to find a more "workable" ...
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2answers
44 views

Conditional Probability in Poker

I'm thinking of a ten person Texas hold'em game. Each person is dealt 2 cards at the start of the game. The question is: GIVEN that you have been dealt 2 hearts (Event B), what is the probability ...
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1answer
50 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...
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2answers
36 views

c(n,k) equals subdivisions

To compare n files, the total comparison count is: $$ {{n}\choose{k}} = C^k_n = \dfrac{n!}{k! ( n - k )!} $$ with k = 2. Input space is composed by all pairs of files to compare. I want to split ...
3
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7answers
162 views

For what $n$ is $n! = 2^8\cdot3^4\cdot5^2\cdot7$?

How can one find $n$ when $n! = 2^8\cdot3^4\cdot5^2\cdot7$? And generally, How to solve this kind of questions? The textbook provided a poor answer.
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1answer
33 views

How do I prove this by induction? [duplicate]

thank you for taking the time to help me with the question. I am struggling to use proof by induction for this formula: $$\sum_{k=0}^{n}k\times k! = (n + 1)! - 1$$ So far, I came up with: $$S(n) = ...
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5answers
293 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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0answers
31 views

Additive and Multiplicative Error in $n!$ Approximation

Let $S(n)=\sqrt{2\pi n}\big(\frac{n}{e}\big)^n$ be the approximation of interest to $n!$. What are good lower and upper bounds on the following two functions $$(1)\mbox{ }|S(n)-n!|?$$ $$(2)\mbox{ ...
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2answers
37 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...
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0answers
25 views

is there anyone able to develop this series in order to get the following equality?

$\sum_{i=1}^\infty (1-\alpha)_{(i-1)}*\frac{\varepsilon^i}{i!}$ = $\frac{1-(1-\varepsilon)^{\alpha}}{\alpha}$ where $(1-\alpha)_{(i-1)}$ is the Pochammer symbol or rising\ascending factorial. Can ...
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3answers
82 views

clarification on the formula $\frac{n!}{(n-k)!}$

$\dfrac{n!}{(n-k)!}$ is used in order to find non-repetitive lists of length $k$ given $n$ possible symbols. For example: find the number of non-repetitive lists of length five that can be made ...
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1answer
38 views

Inequality between factorial and exponential

Trying to find a nice way to simplify the question: Which is bigger 2000! or 1000^2000? I don't know what kind of reasoning I can apply here.
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1answer
134 views

Find the GCD and LCM of the factorials of two given numbers

Find $\gcd(20!, 12!)$ and $\text{lcm}(20!, 12!)$. My answer is: $20=2^2 \times 5$ $12=2^2 \times 3$ GCD $= 2^2 = 4$ LCM $= 2^2 \times 3 \times 5 = 60$ .... But my teacher said that this symbol ...
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0answers
37 views

Determine the formula for hexagon arrangements.

The puzzle to be solved is similar to a jigsaw but using n regular hexagons of equal size for pieces. The pieces are to be placed within a defined perimeter to create a picture. Q: If we let the ...
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2answers
47 views

Factorial simplificaton involving negative 1

What is the best way of simplifying $$\dfrac{(a+b-1)!}{(b-1)!}$$ Ideally i want to get rid of the two $-1$ and the final solution should not containt the gamma function
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1answer
91 views

How to prove that nth differences of a sequence of nth powers would be a sequence of n!

Given an infinite sequence of numbers, first differences denote a sequence of numbers that are pairwise differences, second differences denote a new sequence of pairwise differences of this sequence, ...
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0answers
34 views

How to find a distinct digits appearance from a factorial number?

I want to find out how many times a single digit is appeared in a factorial number. For example- 9! = 362880. There are two times the digit 8 appeared. Again, 13! = 6227020800. Here the digit 2 ...
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1answer
44 views

Bell number vs Factotial

We have $B_n$ is Bell number and $n!$ - factorial. So, what is greater: $n!$ or $B_n$ ? How it can be proven?
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1answer
36 views

Limit of factorials: $\lim_{n\to \infty}\frac {(2n-1)!}{2^{2n-1}n!(n-1)!}$

I'm failing go figure out how to calculate the limit where I have one factorial divided by two at about half its size. The specific limit I'm trying to find is this: $$\lim_{n\to \infty}\frac ...
2
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0answers
91 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
1
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2answers
69 views

for all positive integers m there exists consecutive primes which are at least m apart

I'm having difficulty as to how I should approach this problem, any help would me much appreciated! Note that $k$ divides $n! + k$ for each $k\le n$. Use this fact to show that for all positive ...
1
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1answer
43 views

Prove that if $p\le n$, then $p$ does not divide $n! + 1$

I'm having trouble on how to approach this problem Prove that if $p\le n$, then $p$ does not divide $n! + 1$ ($p$ is prime and $n$ is an integer).
2
votes
3answers
750 views

What is the practical application of factorials

I'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items ...