Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

The factorial is defined as the product of all positive integers less than or equal to some integer $n$, written $n!$. Multiple $!$'s skip integers, so for example $10!!!=10\cdot7\cdot4\cdot1$.

This function is only defined over integers, but the extends it to all complex numbers that are not non-positive integers.

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Does the sequence $\frac{n!}{2^n}$ converge or diverge?

Does the following sequence $\{a_n\}$ converge or diverge? $$a_n=\dfrac{n!}{2^n}$$
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Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
dariusc
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Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
Emil
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An inequality involving factorials and two variables

The problem is as follows: For $m\ge n>1$ prove that $$(m-2)!(n-1)+(n-2)!(m-1)+(m-2)(n-2)\ge (m-1)(n-1)$$ Since $(m-1)(n-1)-(m-2)(n-2)=m+n-3$ so we only need to show that $$(m-2)!(n-1)+(n-2)!(m-1)\ge m+n-3$$. On the face of it this seems to hold but…
Shahab
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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 factorials that isn't something like $\large…
porridgemathematics
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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 Knuth arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)\\f(x,n)=x\underbrace{!\dots!}_n$$ Where there are $n$ factorials…
Simply Beautiful Art
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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 this sum is divided by $120$. Does it divide evenly?…
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How to decide if a factorial is a multiple of certain number?

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?
coder
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Formula to calculate the number of possible positions for $x$ numbers

What formula do I use to calculate the number of possible positions for $x$ numbers? Let's say I have $3$ people in a race. What are all the possible combinations of the order they can finish in? Let's assume ties are not possible. I heard I use…
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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 to start.
beautyofmath
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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? $$n^{(n+1)}(n-1)^{(n+1)+n}(n-2)^{(n+1)+n+(n-1)}\cdot\cdot\cdot 1^{(n+1)+n+(n-1)+\cdot\cdot\cdot+2}$$ Fun fact: The following is in fact…
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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 to writing out every single number on the top…
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Use mathematical induction to prove the following $n! < n^n$

So I'm reviewing some problems but I can't seem to understand the part below, doesn't really have to do with induction but just so you guys understand whats going on. Use mathematical induction to prove the following statement is true for integers…
RiGid
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$t > 2n^2 \implies t!>n^t$ for $n,t \in \mathbb{N}$

I have come across this in a proof: If $t>2n^2$ then, $$t!>(n^2)^{t-n^2}=n^tn^{t-2n^2}>n^t$$ Obviously, this is much help to determine the relationship between factorials and exponential, but I fail to see the motivation behind the initial…
zzzzzzzzzzz
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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 their hat back if, a: No one gets the right kind of…
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