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.

3
votes
1answer
81 views

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, ...
2
votes
1answer
46 views

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.
0
votes
1answer
43 views

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 ...
8
votes
1answer
58 views

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 ...
1
vote
2answers
57 views

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 ...
-1
votes
2answers
65 views

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 ...
6
votes
1answer
136 views

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.
0
votes
1answer
37 views

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 ...
0
votes
3answers
87 views

How to calculate $\lim\limits_{x\to\infty} \frac{\log(x!)}{x\log(x)}$ [duplicate]

How to calculate $\lim\limits_{x\to\infty} \frac{\log(x!)}{x\log(x)}$. Assume base $e$ (so $\ln)$. My attempt: $$\lim_{x\to\infty} \frac{\log(x!)}{x\log(x)}=\lim_{x\to\infty}\frac{\log(1\cdot 2\...
3
votes
2answers
74 views

What is the maximum value of $n$ if $4^n$ divides $1000!$ without a remainder?

If $1000!$ is divided by $4^n$ with a remainder 0, what is the highest possible value of $n$? I placed 2, 3, 4, etc value in $n$ but didn't found any possible $4^n$. Moreover I have seen that only $...
1
vote
2answers
49 views

Prove the following result

Prove that if $p$ is a prime number, then p divides $\binom{n}{p} − \lfloor\frac{n}{p}\rfloor$, for all $n > p$. (where the $\lfloor\frac{n}{p}\rfloor$ denotes the greatest integer less than or ...
0
votes
2answers
37 views

How do I solve this combinatorial proof involving factorial (n)_k?

Let $n$ and $k$ be positive integers with $n \ge k$. Give a combinatorial proof that $$n_k = (n-1)_k + k(n-1)_{k-1},$$ where $n_k$ is a falling factorial: $n_k$ = $n(n-1)(n-2)\ldots(n-k+1)$. I know ...
0
votes
1answer
20 views

Need Help with Some Advanced Integration By Parts Methods

Note: I am asking this question for someone to check my work for me. The problem started out with me finding z! which is equal to the $\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \...
22
votes
2answers
854 views

Find some digits of $17!$

$17!$ is equal to $$35568x428096y00$$ Both $x$ and $y$, are digits. Find $x$ and $y$. So, $$17!=2^{15}\times 3^6\times 5^3\times 7^2\times 11\times 13\times 17=(2^3\times 5^3)\times 2^{12}\times 3^...
0
votes
1answer
42 views

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 ...
-1
votes
2answers
50 views

Inequality with logarithmic function

Prove the following inequality for any $n ∈ [2;\infty)$: $$ \log_{n!}(\frac{n+1}{2})> \frac{1}{n} $$
0
votes
1answer
45 views

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 ?
1
vote
1answer
85 views

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 ...
1
vote
0answers
28 views

Basic Question for limits and Taylor Expansion

I am trying to solve a question for limits. Is $1^x +2^x + 3^x + .....+ n^x$, a Taylor expansion for something? What is the Taylor expansion for $n! $? The question that I'm trying to solve is:- $$\...
1
vote
0answers
63 views

Bhargava's generalized factorials

Manjul Bhargava had generalized the factorial function in number-theoretic context in this paper. At the end of the paper, he mentions some interesting problems associated with the generalized ...
4
votes
1answer
38 views

Solutions to $1/K!+1/L!+1/M!=1/N!$

Is there more than one solution to $\frac{1}{K!}+\frac{1}{L!}+\frac{1}{M!}=\frac{1}{N!}$ where $K, L, M, N$ are all natural numbers? The one solution i came up with was to assume that $K=L=M$, ...
0
votes
1answer
32 views

Prove that given number is integer [duplicate]

I have bumped into one simple task which I am not able to prove: How to prove, that number $ \frac{1000!}{(100!)^{10}} $ is integer?
0
votes
0answers
27 views

Re-express this term as a binomial constant

How can I express the following constant $$\frac{n!}{q!k!r!(n-q-2k-r)!}$$ in terms of the Binomial constant or the falling factorial?
6
votes
2answers
157 views

Prove that $0!+1! + 2! + 3! + … + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three real number

Let $n$, $p$ and $r$ be three positive integers. Prove that for $n \geqslant 3, r>1$, $$\sum_{k = 0}^{n} k! \neq p^\text{r}$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preaparatory Question) I am not so ...
-1
votes
1answer
26 views

Solutions for $n$? Use Stirling approximations if needed

$$(2n)! = a^{2n}$$ where $a \in \mathbb R$, and $n \in \mathbb N$. This is relevant because of a research question I'd asked and received an answer to by Sotiris here
1
vote
2answers
53 views

What's $\frac{(2n)!}{n!}$ equal to?

I chanced upon this problem: $\textbf{Show that} \hspace{0.2cm}\frac{(2n)!}{n!} = 2^n(1 \times 3 \times 5 \times ... \times (2n - 1)).$ I tried the following, and realised I was wrong! : $\frac{(...
1
vote
1answer
19 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 $$ ...
0
votes
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\...
8
votes
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 ...
2
votes
2answers
66 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 ...
0
votes
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 ...
0
votes
3answers
51 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 ...
0
votes
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?
0
votes
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?
0
votes
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 ...
1
vote
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.
1
vote
0answers
43 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 ...
1
vote
1answer
86 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 ...
0
votes
0answers
60 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 ...
2
votes
1answer
44 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 ...
0
votes
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/...
0
votes
2answers
60 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 ...
0
votes
2answers
50 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 ...
0
votes
3answers
85 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\...
-3
votes
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 \\ & ...
0
votes
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, ...
4
votes
4answers
432 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 ...
0
votes
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, ...
1
vote
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 ...
5
votes
0answers
199 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 ...