# 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.

2,472 questions
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### Quotient of factorials

Prove that $\dfrac{(n^2)!}{(n!)^{n+1}}$ is an integer, where $n$ is a natural number greater than $5$. I know how the product of $r$ consecutive numbers is divisible by $r!$ Could we use it here? If ...
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### Recurrence with non-fixed number of variables

Given the recurrence $\forall \{a_i\}\subseteq \mathbb N.\ \ T(\sum_i{a_i},m)\geq \sum_i {T(a_i, m-\prod_i (a_i!))}$ with $T(\Theta(1))=\Theta(1)$ How should I solve this? I tried to minimize the ...
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### Factor of Kaprekar number

I'm trying to get the factor of Kaprekar Number, i.e: In range $1 \rightarrow 100$ there are $1, 9, 45, 55, 99$, so instead of checking all 100 number is Kaprekar Number or not, I'm trying to know ...
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### Approximation $\ln \frac{(h+f)!}{(h-g)!}$ using Stirling’s when $f+g=o(h)$

Stirling’s approximation can be extended to a very well known inequality - $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \leq n! \leq e\sqrt{n}\left(\frac{n}{e}\right)^n$$ How can we use this to prove, ...
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### Factorial function intersections

How would you find the intersection between a factorial function and exponential function? ie work out: $x! = 2^x$ Is it even possible ? As factorials only really work with integer values. When I ...
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### Proof that $\sum\limits_{i=1}^\infty \frac{i}{(i+1)!} = 1$

I came across this result randomly and am quite sure it's right. Is there any way to prove it rigorously? The numerator always seems to be one less than the denominator. Thanks!
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### Attempted algorithm to find which shortest permutation of a string out of “hard.”

Recently, I have been contemplating on how to find an unknown factorial. To find a particular string of text. Update- I removed pi. The formula Z is defined as L= length of string in character ...
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### Show $\frac{ (n+k)!}{n! \sqrt{n+k}} \le ( f(n) )^k \sqrt{k!}$ for some function $f$

Let $k$ and $n$ be positive integers. Can we show the following inequality: \begin{align} \frac{ (n+k)!}{n! \sqrt{n+k}} \le ( f(n) )^k \sqrt{k!}, \end{align} where $f(n)$ is some funciton of $n$...
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### Verification of $(-\frac{1}{2})!$

I was working on a proof for $(-\frac{1}{2})!$ and my issue was with converting my bounds from variable to variable. As you will see, I kept my bounds in terms of $t$ throughout the calculation, but ...
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### Partial sum formula and indefinite integral of the following factorial/Lambert reciprocal function?

function is as follows: $$\frac 1{(2m)!(1-q^{2m})}$$ $$s.t. (m, q) \in \mathbb N$$ I ask this because of some difficulties encountered in a previous question I'd asked, linked here: Previous ...
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### How to find summation of factorials

I got stuck at the following summation while solving another problem. $$\sum_{k=n}^N \frac{(k)!}{(k-n)!}$$ I expanded the summation but have no clue how to simplify it.
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### Calculator giving weird answer when dividing factorial

I am using a TI-34 MultiView I was trying to divide the following 20!/(17!3!) The answer should be 1140 right? the numerator is 2.43*10^18 the denominator ...
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### Similarity between $e^x$ power series and Gamma function integral?

The power series for $e^x$ is as follows. $$e^{x} =\sum ^{\infty }_{n=0}\frac{x^{n}}{n!}$$ If we define $n! = \Gamma(n+1)$, then we have $$n!=\int ^{\infty }_{0} x^{n} e^{-x} dx.$$ An extremely ...
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### Permutation: How to arrange 12 people around a table for 7?

I want to understand how to arrange $12$ people around a circular table with $7$ chairs. We don't care about the overflow, those people can go to another table. I thought the way to solve the problem ...
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### Show that $x! y! = z!$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$.) [closed]

Show that $$x! ·y! = z!$$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$) I am stuck on this problem. Within this section we are learning Congruence. So I know it involves ...
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### How to solve $(x!)!+x!+x=x^{x!}$

How to solve this equation $$(x!)!+x!+x=x^{x!}$$ The answer is $3$ . But I have no idea of how to solve it. Thanks for your time.
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### Analytically continuing the product of the first $n!$ to negative numbers?

Analytically continuing the product of the first $n!$ I recently had the following idea to use the below identity: $$(1!2! 3! \dots n!) (12^2 3^3 4^4 \dots n^n) = n!^{n+1}$$ If we focus on the ...
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### Lottery odds question

In this lottery 5 balls are chosen from 1-50. A friend offers me 5 to 1 if I can get one number correct from the 5 chosen, the order it comes up doesn't matter. Is this a good bet to take? To get ...
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### Inequality with logarithmic function

Prove the following inequality for any $n ∈ [2;\infty)$: $$\log_{n!}(\frac{n+1}{2})> \frac{1}{n}$$
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### What are the steps to solve this problem? [closed]

If $\frac{1}{2n!}$ , $(n-2)!$, $(2-n)!$ are the side lengths of a triangle in cm., then what is the numerical value of the area of the triangle ?
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### A subword in a word

Probability So I have been trying to solve this question in probability, but I don't seem to get the correct answer. I am not bad at probability and this seems to be easy one, but I'm just struggling ...
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