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.

2,555 questions
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Sum of product of Stirling numbers

We have for $n>0$, $k>0$ $$\sum\limits_{j=1}^{\min(n,k)}(j!)^2{n\brace j}{k+1\brace j+1}=\sum\limits_{j=0}^{\min(n,k-1)}j!(j+1)!{n+1\brace j+1}{k\brace j+1}$$ How can we prove it?
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Calculating factorial in terms of certain powers

Factorial $n!$ is defined to be$$n!=1·2·3·4\cdots n$$Let's choose base $2$ to represent $n!$, thus$$n!=2^0·2^1·(2^1+2^0)·2^2·(2^2+2^0)·(2^2+2^1)·(2^2+2^1+2^0)\cdots$$Of course, it is just binary ...
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Proving the summation of a double factorial infinite series.

$$\sum_{n=0}^{\infty }\frac{(-1)^{n}((2n-1)!!)^2}{(2n)! (2^{2n})} = \frac{2}{\sqrt{5}}$$ I came across this summation through some other work, came across the solution as part of a function, but I ...
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Remainder theorem of factorials and powers

I took a competitive exam and there was this question If $7^n$ divides $68!$ Then what is the greatest value of $n$? Can someone tell me how to find the answer.
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Confusion regarding the scope of multiplying a factorial by a constant $30(3k)!$

I've stumbled upon this factorial, which is written exactly in this way: $$30(3k)!$$ As simple as this may seem, I honestly don't know how to interpret this. Does the factorial encompass the entire ...
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Find value of n if ${(n-1)! \over (n-3)!} = 30$

So here's my attempt : $${(n-1)(n-2)(n-3)! \over (n-3)!} = 30$$ $${(n-1)(n-2)} = 30$$ then $$n-1 = 30$$ or $$n-2 = 30$$ then $$n = 31$$ or $$n = 32$$ but my textbook says $$(n-1)(n-2) = 6 * 5$$ ...
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Scary looking limit with an elegant answer

This problem was posted half a year ago by Pierre Mounir on a Facebook group and until now it received no answers. Since most of his problems that I saw were amazing I can bet this one it's worth the ...
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Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)!$$ ...
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Determine $N$ of series $\sum_{n=1}^{N}\frac{n^n}{(2n+1)!}$ so that it differs from the actual sum by less than $\frac{1}{200}$

I can establish that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{n!}$$ But $$\sum_{n=1}^{+\infty}\frac{n^n}{n!}$$ diverges (by the ratio test). And even if it converged I wouldn't have ...
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Distribute 14 books to 2 people so that each has at least 3 books.

I've been trying to solve a problem, but the solution I got looks extrmely unlikely to be right. Perhaps some can point where I'm wrong. The problem is the following. We have $14$ different books. We ...
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In how many ways can 7 women, 10 men sit at table such that no woman sits besides another?

We have $7$ women and $10$ men; they sit at a table. I've been trying to solve how many ways can they sit excluding the case of women sitting next to each other. My reasoning was the following: I ...
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Given positive integers a and b, find values for a and b if a! * b! = a! + b! [closed]

Given positive integers a and b, find values for a and b if a! * b! = a! + b! I have no clue where to start and would appreciate some help, thanks!
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How to show that intermediate values of $\binom{n}{\lfloor n/2 \rfloor }$ are all integers that does not exceed $\binom{n}{\lfloor n/2 \rfloor }$

Let n be a positive integer, and assume that j is a positive integer not exceeding $n/2$. Show that in $n \cdot (n-1)\cdot (n-2)\dotsm(n-j+1)\,/\,j!$, if one alternates the multiplications and ...
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Why $\lim_{n\rightarrow \infty} \frac{n!}{n^{k}(n-k)! } =1$?

I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable. Consider the binomial ...
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Recently I have been creating a program that uses Wilson's theorem to solve for prime numbers, but the main disadvantage of that system is that most computers have the 64 long limit, which comes out ...
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The behaviour of $\operatorname{Im}(!n)$

What's going on with the behaviour of the subfactorial's imaginary part? Background: Out of curiosity I tried to construct some recurrence relations using the Pochhammer symbol and out of those came ...
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How limitations affect the number of possible pandigital numbers

First, I understand that the definition of "pandigital" can vary, somewhat, so for the purposes of this question, here's the definition we'll use: "Positive and whole numbers that include each ...
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Solution of $n!=p+1$ with $p$ is prime number?

One of my friend asked me to solve this equation $n!=p+1$ with $p$ is prime number and n is positive integer , it's clear that for $p=2$ there is no solutions because : $n! < 3$ for $n=1$ , But ...
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Prove that $n$ factorial is larger than $n$ cube if $n$ is large enough

I try to solve it by Mathematical Induction. However I don't know how to prove that $(k+1)!>(k+1)^3$
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How can you further simplify a subtraction of two large factorials [closed]

100!-99! I am having a bit of difficulty with this question. Thanks in advance
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Ratio of large binomials in matlab

I need to compute the following ratio $$\frac{n!}{j!(n-j)!}/ \frac{n!}{(n/2)!(n/2)!}$$ I've tried to do this using nchoosek which works finte until $n\approx1000$. But I need at least $n\approx 10000$...
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$\lim_{x \to \infty} {{(x!)^p}\over {x^x}}$

After some experimentation, I am pretty sure that $$\lim_{x \to \infty} {{(x!)^p}\over {x^x}} \to \infty$$ for $p \gt 1$, and that $$\lim_{x \to \infty} {{(x!)^p}\over {x^x}} \to 0$$ for $p \le 1$. ...
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Quotient of factorials

Prove that $${(n^2)!\over(n!)^{n+1}}$$ 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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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 $\sum\limits_{i=1}^\infty \frac{i}{(i+1)!} = 1$ [duplicate]

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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Maximising $\frac{x^n}{n!}$

Let $x$ be a number such that $x\gt 0$ and $x\in\mathbb{R}$. Is it true that the maximum value of the expression $$\frac{x^n}{n!}$$ occurs for $n\in\mathbb{N}$ where $n=\lceil x \rceil - 1$? If true, ...
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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 ...
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 ...
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.