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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2answers
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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
46 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
32 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 ...
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7answers
147 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
29 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
153 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
30 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
36 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
22 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
80 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
25 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
54 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
32 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
43 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
33 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
26 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
38 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
28 views

Limit of factorials

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
88 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 ...
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2answers
58 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 ...
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1answer
29 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).
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3answers
212 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 ...
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4answers
98 views

Why is ${{n+1}\choose{k}}={{n}\choose{k-1}}+{{n}\choose{k}}$? [duplicate]

My teacher showed us a proof by induction for this equation for $n\in\mathbb{N}$: $$\sum\limits_{k=0}^n{{n}\choose{k}} = 2^n$$ In the first step, this sum is rewritten using ...
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0answers
41 views

How come negative factorials never give us an answer?

I've done this and it always gave me an error probably because of this (it'll continue):$$4!=4*3*2*1=24$$$$3!={4!\over 4}={24\over 4}=6$$$$2!={3!\over 3}={6\over 3}=2$$$$1!={2!\over 2}={2\over ...
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1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
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4answers
67 views

How can we find factorials in decimal form? [duplicate]

I've heard of factorials such as $5!$ and $3!$, which work like this: $5!=5\times4\times3\times2\times1=120$ and $3!=3\times2\times1=6$. At least this is what we get. Also, surprisingly, $0!=1$, but ...
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3answers
96 views

To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 … (2n-1)\over 2.4.6…(2n)$

How can I show $\lim_{n \to \infty} a_n = 0$ $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$ I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below ...
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10answers
185 views

How to evaluate $\lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$

Im tempted to say that the limit of this sequence is 1 because infinite root of infinite number is close to 1 but maybe Im mising here something? What will be inside the root? This is the sequence: ...
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3answers
54 views

Calculating the nth derivative of $\frac{x}{x+1}$

I was asked to calculate the nth derivative of $f(x) =\frac{x}{x+1}$. My solution: $$ f'(x) = (x+1)^{-2}$$ $$f''(x) = (-2)(x+1)^{-3}$$ $$f'''(x) = (-2)(-3)(x+1)^{-4}$$ $$f^{n}(x) = n!(x+1)^{-(n+1)} . ...
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3answers
38 views

factorial division when the bottom number is larger than the top number

I have a factorials problem to solve, and I do not know the method of solving it. I know how to do one number factorials (e.g. 5!, 15! etc...) and factorial division where the top number is larger ...
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2answers
88 views

Prove that $\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+a)}=\frac{1}{aa!}$ [duplicate]

$$\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+v)}=\frac{1}{vv!}$$ I am struggling to find a solution for this but no luck yet. How can I analyze it to get to second part?
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1answer
29 views

Factorial Divides Rising Power Proof Help

I'm trying to prove the following: $m^{\overline n} \equiv 0 \bmod n!$ Where $m^{\overline n} = m\left({m+1}\right)\left({m+2}\right)\ldots\left({m+n-1}\right)$, the product of $n$ successive ...
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0answers
58 views

Relation between Hyperfactorial, Superfactorial, Pascal's Triangle and Binomial Coefficient

I read here that the product of the elements in the $N^{th}$ row of Pascal's triangle is equal to $(n!)^{n+1}/(\prod_{k=1}^n k!)^2$. Let's call the product of elements in the $i^{th}$ row of Pascal's ...
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1answer
37 views

Rising factorial power

How the expression below can ve proved: $(a + b)^{\overline{n}} = \sum\limits_{j=0}^{n}C_n^j a^{\overline{n-j}}b^{\overline{j}}$ where $x^{\overline{n}}$ - is rising factorial power: ...
5
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2answers
145 views

How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)…(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$

Does anyone have any idea how to prove that $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$$
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4answers
89 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
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0answers
40 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
11
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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 $p$ ...
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4answers
154 views

Is this a demonstration or a definition?

Some people says that this can be demonstrated $0!=1$, but other say that this is a definition. Which one is correct? Let's given $n\in\mathbb{N}$: $$(n+1)!=(n+1)\cdot n!$$ $$(0+1)!=(0+1)\cdot 0!$$ ...
4
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1answer
92 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
4
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2answers
50 views

Is there a Sum Factorial?

I am curious if there is any addition factorial. Obviously, $$x! = \prod_{n=1}^x n$$ but what I want is a shorthand way of writing: $$\sum_{n=1}^x n$$ So is there such a thing? and if so, what is ...
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2answers
87 views

How to calculate decimal factorials, like $0.78!$ [duplicate]

When I enter $0.78!$ in Google, it gives me $0.926227306$. I do understand that $n! = 1\cdot2\cdot3 \cdots(n-1)\cdot n$, but not when $n$ is a decimal. I also have seen that $0.5!=\frac12\sqrt{\pi}$. ...
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0answers
86 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + ...
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4answers
35 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
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8answers
212 views

Prove that $n! > n^5$

I'm trying to prove that $n! > n^5$ for large enough values of n. While it seems obvious that this should be true, I have no idea how to prove it rigorously. EDIT: So, looking at the comments, ...
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4answers
84 views

How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
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1answer
37 views

Factorials with fractions

I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$ becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$ and then how it becomes $$ \frac {n+n^2+n}{n-1} $$ I've tried applying ...
0
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2answers
94 views

How to evaluate factorials greater than $69!$

How to evaluate factorials greater than $69!$? On my calculator, $69!$ is the largest number I can enter before it gives me a syntax error, most likely due to an overflow. Is there a way to evaluate ...
4
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1answer
160 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
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3answers
91 views

Could negative integer factorials be defined in some way?

I know that, calling $F$ such an extension, if we wanted to keep having $$ F(z+1)=(z+1)F(z),$$ letting $ z=-1$ would lead to the absurdity $ 1=0$. Also, $ \Gamma(z)$ has poles at $ z \in ...