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.

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1answer
20 views

recursive relation on derangement of objects

Let $a_{n}$ represent the number of derangements of $n$ objects . If $a_{n+2}=p a_{n+1}+q a_{n}\;\forall n\in\mathbb{N}$ then what is $\displaystyle \frac{q}{p}$? What I have tried: I have used $$ ...
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1answer
37 views

Finding the last non-zero digit of $n!$ in $O(1)$

I saw a few approaches of finding the last non-zero digit using recurrence relation, CRT etc. I came up with a trivial $O(1)$ approach but didn't find it anywhere so asking it here. We can write $1\...
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1answer
97 views

$N! \pmod{P}$ (huge numbers)

What is the value of $2019! \pmod{7}$? I guess it's $0$? Because $$2019! = 2019\cdot2018\cdot2017\cdot ...\cdot7\cdot6\cdot...\cdot1$$ There's $7$ and also numbers that has $0$ remainder when divided ...
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2answers
70 views

can we perform modulo operator on a fraction on both of it's numerator and denominator?

I want to calculate nCr (mod $10^9+1)$.so for calculating nCr we have: $$nCr=\frac{n!}{r!(n-r)!}$$ so I want to know whether it is true that I perform modulo operator to numerator and denominator ...
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2answers
38 views

How to complete this proof involving factorials

Recently I came across the following identity, but if I try proving it with induction, then I get stuck. $$n! = \sum^n_{k=0}(-1)^{n-k}\binom{n}{k}(k+1)^n$$ While trying my induction step I get the ...
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3answers
54 views

A lower bound for de Polignac's formula

De Polignac's Formula has many uses, for example when calculating the number of trailing zeroes of $n!$ :$$\nu_5(n!)=\sum_{i\le\lfloor\log_5n\rfloor}\left\lfloor\frac n{5^i}\right\rfloor.$$ For the ...
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1answer
42 views

Does $N! = 2^m$ hold for any integer values of $N$ and $m$?

For any value of $N$, is it possible that the factorial of $N$ is equal to a power of 2?
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1answer
27 views

Proof for $\sum_{r=1}^{n}r(r!)=\sum_{r=1}^{n}[(r+1)!-r!]=(n+1)!-1$ [duplicate]

I came across the form $\sum_{r=1}^{n}r(r!)=\sum_{r=1}^{n}[(r+1)!-r!]=(n+1)!-1$ while solving a question in determinants. How do we get to the formula stated above?
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1answer
51 views

Storing numbers in an efficient way in computer.

I know that we can write a very large number such as 5040 in only 7!, and imagine I want to store this number in a binary file with the least number of bits. Saving 5040 takes 13 bits of space, while ...
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1answer
104 views

Evaluate the following limit: $\lim\limits_{ n\to\infty}\frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$

Evaluate $\lim_{n \to +\infty} \frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$. Please help with steps, Dont know how to break it down to cancel out terms.
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0answers
44 views

How to find the nearest factorial to a number

How can I find the nearest Factorial to a number? For example I know that the nearest factorial to 200 is 5! So how can I also ...
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1answer
93 views

Double-precision algorithm for inverse log gamma or log factorial?

Question in a nutshell: Can anyone point me to an algorithm for computing to double-precision floating-point (roughly 16 digits) the inverse of either log gamma or log factorial? In other words, if ...
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0answers
72 views

Counting number of occurrences of a number in a factorial

Consider that I want to count the number of times 360 occurs in 520! $360 = 2^3 \cdot 3^2 \cdot 5^1$ $520! = 1\cdot2\cdot3\cdot4\cdot\cdots$ As it can be noticed, $2$ occurs at least $3$ times ...
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1answer
48 views

Help with proving an equation factorial-time complexity

I've been recently asked by one of my friends to prove an equation but still, I'm confused how to get it started tho. log(n!)= θ(nlog(n)) Does anyone know how to help? I'll be very grateful if ...
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0answers
18 views

Reducing a large number to a smaller number using the Factorial Number System

Good day all, I have a large number 373335438 that I would want to reduce to a smaller number using the Facorial Number System here https://en.wikipedia.org/wiki/...
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2answers
69 views

Finding an Inverse of Restricted Gamma Function

I don't know/haven't used LaTeX yet but I'll do my best to keep it simple, I'm working on my undergrad senior project and I'm trying to find an inverse function for f(x)=(x-1)! just in the positive ...
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2answers
58 views

How do I find sum of digits of a given factorial with missing digits?

Suppose its given that 21!=5109094x17170y440000 How do I find x+y I know any factorial bigger than 6! will be divisible by 9. So I can apply that rule to find out it should be 52+x+y ...
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3answers
86 views

How to calculate 15! without using calculator

I am joining a maths competition and recently I am preparing for it. I came across a question that asks me to fill the blank of a number: 1_0767436_000 And this number is the product of $15!= 15\...
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2answers
78 views

prove that [p!/(p-4)!] + 1 is a perfect square for all natural p.

one can observe that $[p!/(p-4)!] + 1$ is basically the product of four consecutive integers plus one.Since this is $$ \begin{eqnarray} p(p+1)(p+2)(p+3)+1 & = &(p^2+3p)(p^2+3p+2)+1 \\ & ...
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0answers
11 views

Closed form of basic hypergeometric series: Adapt the answer given in linked question to solve my own variation?

I'm interested in two things: A computationally efficient (used here to mean the number of terms is bounded and not dependant on the size of the input) partial sum formulae of the expression below, ...
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4answers
435 views

How to prove $ \frac{m!}{n!} \geq n^{m-n} $

How to prove the following: $$ \frac{m!}{n!} \geq n^{m-n} $$ In my book it's written: "easy to prove by separately considering the cases $m \geq n$ and $m<n$). I tried using the bounds of ...
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1answer
22 views

Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $\zeta(-z)$=${-2*z!\over(2\pi)^{z+1}}$$sin({\pi z\over2})$$\zeta(z+1)$This formulas expresses $\zeta(-z)$ in terms of $\zeta(z+1)$ Note: I read that the author said, ...
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1answer
58 views

Can the sum of the first $p$ factorials ever be a perfect power for $\ p>3\ $?

Has $$\sum_{j=1}^p j!=q^r$$ , where q,p,r are positive integers, and r > 1 , a solution ? I solved partially, if r is even, then RHS is a perfect square, and there is no doubt in that. Therefore, the ...
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0answers
201 views

Finding sum to infinity: $\sum\limits_{n = 1}^{ \infty}\frac{n^2}{n!}$ [duplicate]

I am trying to find what this value will converge to $$\sum_{n = 1}^{ \infty}\frac{n^2}{n!}$$ I tried using the Taylor series for $e^x$ but couldn’t figure out how to manipulate it to get the above ...
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1answer
78 views

$(\frac{n}{e})^{n} < n! < (\frac{n}{e} + n\varepsilon)^{n}$ doesn't comply with the limit definition?

I try to understand what I've overlooked, when I came up with this inequality: First, we have this limit: $$\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} = \frac{1}{e}$$ Which gives, by the ...
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3answers
115 views

Longest sequence of consecutive integers which are not coprime with $n!$

For any integer $n$, the factorial $n!$ is the product of all positive integers up to and including $n$. Then in the sequence $$n!+2,n!+3,... ,n!+n$$ the first term is divisible by $2$, the second ...
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2answers
58 views

Check if $k$ is divisible by $2^9$ or $2^{10}$

$k=\frac{512!}{256!*128!*...*2!*1! } $ I need to check if the expression k is divisible by $2^9$ or $2^{10}$. This is a multiple choice question and the options and the question goes like this: ...
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2answers
39 views

Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N??

Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N? where a is an arbitrary integer, and N is a prime. e.g. i know 22! mod 23 ≡ 22 using Wilson's theorem. lets say i want to know ...
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3answers
71 views

Can't find a seemingly simple limit $\lim_{n\to\infty}\frac{(n+k)!}{n^n}$

Evaluate the limit: $$ \lim_{n\to\infty}\frac{(n+k)!}{n^n}, \ n,k\in\Bbb N $$ I would like to avoid Stirling's approximation, derivatives and Cesaro-Stolz, since none of them has been yet ...
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0answers
45 views

A question on the approximation of $\ln(x!)$ for small x

In a question of Arfken and Weber's Mathematical Methods for Physicists, For small values of $x$, $\ln(x!)=-\gamma x+\sum\limits_{n=2}^{\infty}(-1)^n\frac{\zeta(n)}{n}x^n$, where the symbols used ...
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2answers
87 views

Show that $\frac{(2n)!!}{(2n+1)!!}$ converges.

Given a sequence $\{x_n\}, \ n\in\Bbb N$: $$ x_n = \frac{(2n)!!}{(2n+1)!!} $$ Show that $x_n$ converges. I'm wondering why I'm getting a seemingly wrong result (assuming the problem statement ...
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6answers
136 views

Determining the missing digits of $15! \equiv 1\square0767436\square000$ without actually calculating the factorial

$$15! \equiv 1\cdot 2\cdot 3\cdot\,\cdots\,\cdot 15 \equiv 1\square0767436\square000$$ Using a calculator, I know that the answer is $3$ and $8$, but I know that the answer can be calculated by ...
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1answer
128 views

Methods for proving a function outputs an infinite number of integers

I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in ...
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2answers
79 views

Probability an ace lies behind first ace

Consider a deck of 52 cards. I keep drawing until the first ace appears. I wish to find the probability that the card after is an ace. Now, the method I know leads to the correct answer is that given ...
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1answer
98 views

Calculate $\sum_{k=l}^{2n} \binom{2n+k}{2k} \frac{(2k-1)!!}{(k-l)!} (-1)^k$ [closed]

I'm trying to proof that $$\sum_{k=l}^{2n} \binom{2n+k}{2k} \frac{(2k-1)!!}{(k-l)!} (-1)^k = \begin{cases} 0 \quad {\rm if} \, \, l \, \,{\rm odd} \\ \frac{(-1)^{n-l/2}(2n+l)!}{4^n \left(n-\frac{l}{2}\...
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1answer
36 views

Need help understanding a step in an induction proof

I want to prove that $(n!)^{(n-1)!}$ divides $ n!!$ via induction. I was going through a post I found on Quora doing this, but I got hung up on the last step. For the sake of legibility I'll rewrite ...
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2answers
45 views

Further explanation for steps of an equation that proofs that $\sum^{n}_{k=0}k\cdot \binom{n}{k}=n\cdot2^{n-1}$

So, this is one of the questions in my textbook, which seems to be quite common: $$\sum^{n}_{k=0}k\cdot \binom{n}{k}=n\cdot2^{n-1}$$ The same book provides the following solution: $$\sum_{k=0}^{n}k\...
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2answers
45 views

why is $\frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$?

This is an equation from my textbook that I am trying to understand: $$ \frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$$ What I got so far, is that $\frac{k\cdot n!}{k!} = \frac{...
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2answers
53 views

Turning product sequences into factorials

I am trying to figure out the steps between these equal expressions in order to get a more general understanding of product sequences: $$\prod_{k=0}^{n}\left(3n-k\right) + \prod_{k=n}^{2n-3}\left(2n-k\...
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2answers
41 views

Simplyfying factorials why is $(n+1)(n-1)!-(n-1)! = n(n-1)!$?

I am a bit unclear on how these two expressions are equal: $$(n+1)(n-1)!-(n-1)! = n(n-1)!$$ So far, I obtained $$\frac{(n+1)!}{n}-(n-1)!=\frac{(n+1)n(n-1)!}{n}-(n-1)! = (n+1)(n-1)!-(n-1)!$$ ...
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2answers
54 views

What would be the sequence of $n!!$ (double factorial)

I know that the sequence of $n!$ is $$n(n-1)(n-2)\cdots(2)(1)$$ but what would be the sequence of $n!!$? (In the interest of clarity, this is also known as the double factorial, not to be confused ...
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3answers
72 views

How find the limit of $\lim\limits_{n\to \infty }\left(n-1\right)!$

I have the next limit: $$\lim\limits_{n\to \infty }\left(\sqrt[n]{\left(\frac{n!-1}{n!+1}\right)^{\left(n+1\right)!}}\right)$$ I had done some steps and simplified it to: $$\lim\limits_{n\to \infty ...
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3answers
174 views

Is it possibile to obtain the sum of the following series without using hypergeometric functions?

I know that the following series: $$ \sum_{n=1}^{+\infty}\frac{ (n!)^2}{(2n)!} $$ converges. If I plug it in Wolphram Alpha, I can see that its sum is $$ \frac{1}{27} \left(9 + 2 \sqrt{3} \pi\right). $...
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3answers
41 views

Prove $2\times1!+5\times2!+10\times3!+…+(n^2+1)n!=n(n+1)!$ for all positive integers

I am trying to prove by mathematical induction $2\times1!+5\times2!+10\times3!+...+(n^2+1)n!=n(n+1)!$ for all positive integers $n$. So far I have: Solved in the first case possible - $1$ Assumed ...
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1answer
47 views

Finding Bounds in proof of Stirling's Formula

In a proof of Stirling's Formula, my professor claims that $\frac 12 $($ln$k + $ln$(k+1)) $\le$ $\int_k^{k+1}$$ln$x $dx$ $\le$ $\frac 12$($ln$k + $ln$(k+1)) + $\frac 1{k^2}$. I can see that the ...
2
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3answers
51 views

Number of divisors of $10!$

Determine the amount of divisors of $10!$ This is a question in my combinatorics textbook, so I need to somehow reduce this to an elementary counting problem like combinations, permutations with or ...
1
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5answers
105 views

Show that $\lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty$

Show that $$ \lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty $$ The only way i've been able to show that is using Stirling's approximation: $$ n! \sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n $$ Let: $...
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0answers
32 views

How to get this closed form for such recurrence?

We have for $k>0$, $n>0$, $m\geqslant0$ $$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$ also $$p_0(n,m)=\begin{cases} (n-1)!,&\text{$n>0, m=0$}\\ 0,&...
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0answers
31 views

Intuitive method for solving $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$

Is there any intuitive method of solving: $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$ without having to develop the Exponential Integral Ei?
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2answers
174 views

Geometrical interpretation for the sum of factorial numbers

I am in need of a way to represent the sum $1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$ in a geometrical way. What I mean by this is that for example, the sum $1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 =...