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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33 views

expanding a factorial by simplifying with $(k-1)!$

I'm wandering how $\frac{(k+2)!}{(k-1)!} = \frac{(k+2)(k+1)(k)(k-1)!}{(k-1)!}$. I'm confused about how $(k-1)!$ can be inserted into the numerator here. I know that this eventually works out to $k^3+...
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To where the sum $\sum_{n=2}^N \frac{a^n}{n!}$ converges?

I'm trying to implement this expression in an algorithm, however computing factorial is very time-consuming, so I'm trying to find an equivalent equation that avoids that factorial term. It can be an ...
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28 views

A question on sum the series: [on hold]

I'm not getting right approach: $$\frac{1^2}{3!} + \frac{2^2}{4!} +\frac{3^2}{5!}\ldots$$ till infinite terms.... Answer is given to be : $2e-5$
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1answer
66 views

Are Factorials just Polynomials

I was solving few limit questions based on factorials . They looked scary at first imagining factorials , but you take few terms out and you are done. My question is can we find a polynomial such $${...
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Zeros at the end of sum of factorials

Find the number of zeros at the end of $15! + 16! + 17! + 18!$ ? I know the method find the number of zeros at the end of x! where $x = { 15! , 16! , 17! ...}$ by dividing by number by $5,5^2, 5^3$ ...
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26 views

Number of arrangements of one $1$, two $2$'s, three $3$'s, four $4$'s, …, and nine $9$'s (given constraints)

We have no zeros, one $1$, two $2$'s, three $3$'s, four $4$'s, five $5$'s, six $6$'s, seven $7$'s, eight $8$'s, and nine $9$'s. How many $12$-digit numbers can we make using at least three $6$'s so ...
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Project Euler #239 Java [closed]

A set of disks numbered 1 through n are placed in a line in random order. What is the probability that we have a partial derangement such that exactly k prime number discs are found away from their ...
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50 views

“Non-trivial” solutions to equal products of consecutive integers

$\bullet\ \textbf{Question}$ One can find equivalent products of consecutive integers such as $$8\cdot9\cdot10\cdot11\cdot12\cdot13\cdot14=63\cdot64\cdot65\cdot66.$$ Other solutions of this have been ...
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1answer
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Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,…,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$

I try to prove the following formula: $$n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$$ I noticed that $\upsilon_{p}(\operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)) = s$ iff $\...
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1answer
53 views

A combinatorial question with an easy rolling dice game

Recently, I had played a dice game with my classmates. The game is like that: Find a person rolls a dice and got a number $X$, then he rolls it again. If he gets $X$ again, the game stops; if he ...
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1answer
83 views

Majoring sum with factorials

EDIT: sorry for all the crappy formulas, I tried to add the more context I can, I hope it is better. I am currently trying to analytically bound the following term: $$ \left\vert\left\vert e^{\lambda ...
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1answer
60 views

Find the sum $ 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + … + n \cdot n! $ [duplicate]

Find the sum $$ 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + ... + n \cdot n! $$ Attempt $$ 2 \cdot 2! + 3 \cdot 3! = 2 \cdot 2! + 3^{2} \cdot 2! = 2! (2+3^{2})$$ $$ 4 \cdot 4! + 5 \cdot 5! = 4 \cdot 4!...
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1answer
34 views

Exponent-factorial inequality marathon [closed]

A) I am wondering which one is bigger? $(((5!)!)!)!$ or $5^{5^{5^5}}$. B) And if there is a largest number with at most four $5$’s and four operations, or there is no such number. Here, we define ...
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1answer
34 views

Can we use Bernoulli’s inequality to show that (a^n)/n! goes to zero as n goes to infinity?

From my problem set, I need to show that $(1+a)^n \ge 1+an$ for any $(a, n) \in [-1, \infty) \times \mathbb{N}$ by induction, and use this inequality to show that i) $a^n \rightarrow \infty$ if $a>...
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47 views

How to simplify : $\frac{(x-4)!}{(x-1)!}$ ??

Isn't $(x-4)!$ equal to $(x-4)(x-3)(x-2)(x-1)$? See the picture. How it is equating? Thanks!
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2answers
87 views

How many $0$'s are at the end of $(38!)^{20}$? [closed]

I am getting $160$ as my answer but in the book, it is $168$. Which is the correct answer?
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1answer
173 views

Prove that summation is less than 1

Prove that $m! \times \sum_{n=m+1}^\infty \frac{1}{n!} < 1$ I started proving this by simplifying it to: $= m! \times \left(\frac{1}{(m+1)!} + \frac{1}{(m+2)!} + ...\right)$ $= \left(\frac{1}{(m+1)...
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1answer
23 views

What's the links between factorial,Basel Problem and Euler constant .

I'm studying the factorial function and I have found the following identity : $$\frac{\partial^2 x!}{\partial x^2}=-2\gamma+\gamma^2+\frac{\pi^2}{6}$$ At $x=1$ But the Basel problem says : $$\...
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4answers
79 views

Last non zero digit in 20! [duplicate]

So I have a question where it says to find the last non zero digit of $20!$ I proceeded in the following way: Found the prime factorization of $20!$ by calculation the greatest powers of $2,3,5,7,11,...
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60 views

$\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ [duplicate]

Show that $\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ for $0\le k\le p^{a}-1$. Solution: $\binom{p^a-1}{k}=\frac{(p^a-1)(p^a-2)\cdots(p^a-k)}{k!}$. Idea: Then power of $p$ in $k!$ must be less than ...
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1answer
188 views

What is the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$?

Without using computer programs, can we find the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$? What I know is that the last non-zero ...
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73 views

a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
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137 views

To prove that $(n-1)!+1$ is not a power of $n$.

If $n$ is composite, prove that $(n-1)!+1$ is not a power of $n$. Hint: We know that if $n$ is composite and $n>4$ then $(n-1)!+1$ is divisible by $n$. My Solution: Since $n=4$ is the first ...
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40 views

How do you solve an inequality that involves factorials?

I have this inequality that's part of a larger problem (regarding Taylor expansions). I have the steps to the solution, but I don't understand this part: Step 1: $e/1000 > 1/(n+1)!$ Step 2: $...
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1answer
187 views

The units digit of $1!+2!+3!+4!!+5!!+\dots+k\underset{\left \lfloor \sqrt{k} \right \rfloor \text{ times}}{\underbrace{!!!\dots!}}$

For natural numbers $n\ge m$, let $n\underset{m \text{ times}}{\underbrace{!!!\dots!}}=n(n-m)(n-2m)(n-3m)\dots$ where all factors are natural numbers (we exclude $0$ and negative factors). Question: ...
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1answer
59 views

How to calculate the fraction factorial without doing it manually?

We were asked to list the first $3$ terms of $$\frac1{\sqrt{4+3x}} = \frac12\left(1+\frac{3x}4\right)^{-\frac12}$$ which is straightforward and easy. Using the following with $n=-1/2$: $$(1+a)^n = {...
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1answer
22 views

Using Limits to extend the Factorials to Real numbers (and decimals)

Ref: Euler's limit formula for the factorial function I recently read about the Gamma function which works as a factorial operator for Natural Numbers. Most of the books even consider it as an ...
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1answer
30 views

Showing that the number of queues is uniquely expresssed as the product of $2010$ positive integers.

This problem came from a friend in preparation for a contest. There are $2011$ people in a queue lining up for a conference, and no two people have the same height. Bob is the $27$th tallest person ...
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59 views

Find the desired function or disprove its existence

Let $T(n,m)=\frac { n^2\cdot m\cdot f(n)}{n!}$. I need to find $f$ in terms of $n$, such that: $f$ is non decreasing function $f(n)\in\Omega(1)$ $\exists k>0.\ f(n)\in O(n^k)$ The following ...
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2answers
74 views

$n$-th derivative of $x^\alpha$ where $\alpha = m + 1/2$

It is well-known that, for any real $\alpha$ and nonnegative integer $n$ $$ \frac{d^n x^\alpha}{dx^n} = \alpha(\alpha-1)\cdots(\alpha - n + 1) x^{\alpha - n} $$ I just found out that the coefficient ...
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1answer
117 views

What is longer? Length of $2^n$ or number of $0$ in $n!$?

What is longer? Length of $2^n$ or number of $0$ at the end of $n!$ ? Assume that $n$ is really big, big number... Solution Look at terms of $2^k$: $$2,4,8,16,32,64,128,256,512,1024,2048,4096... $$ ...
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1answer
101 views

Show that $(a!)^b b! \mid (ab)! $ [duplicate]

Show that if $a$ and $b$ are positive integers then $(a!)^b b! \mid (ab)! $ This is what I have done: The above statement is true for $a=1$ and any arbitrary value of $b$, and also for $b=1$ and ...
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3answers
74 views

For What Non-Negative Integer Values of $n$ is $n!\geq 3^n$

How could I solve for the $n$ in this instance using discrete methods or is this something that I have to do by hand/computer? I've seen this problem in inductive proofs but the base case is usually ...
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1answer
38 views

Simplifying an alternating sum of a product of factorials

For integers $a$ and $b$, I am curious how to simplify an expression of the form $$\sum_{k=1}^n (-1)^k (a+k)! (b+k)!$$ I assume there is some simplification using properties of gamma and beta ...
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0answers
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What is the growth rate of a multivariable factorial divided by a multivariable factorial? [closed]

Consider the function $\frac{n!}{(n-2)!}$. The fraction simplifies to $n(n-1) = n^2 - n$, so the function has quadratic growth. Add a new variable into each factorial term: $\frac{(mn)!}{((m-2)(n-2))!...
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143 views

Probability of duplicated games in chess

I am just talking with a chess friend on Lichess about the probability to have duplicate games in chess, and some questions arose for me. Taken that a chess game always lasts $P$ (ply) half moves and ...
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3answers
187 views

Solve $a!b!=a!+b!+c!$ where $a$, $b$ and $c$ are nonnegative integers.

My teacher in Math Team gave the following question to us. Solve $$a!b!=a!+b!+c!$$ where $a$, $b$ and $c$ are nonnegative integers. I found only one solution by trial and error and it is $(a,b,c)=(...
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1answer
45 views

How to prove this factorial equality

During my studies, I've come across this strange relation. $$\sum_{i=0}^{q-j-1}\left(\frac{\left(-1\right)^{i}}{i!\cdot\left(q-i-j-1\right)!\cdot\left(q+i\right)}\right)=\frac{\left(q-1\right)!}{\...
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1answer
23 views

Prove the given inequalities. [duplicate]

Prove the following inequalities,$$ (n!)^2 ≤ n^n(n)! <(2n)!$$ My attempt I proved one of the inequality using mathematical induction. To prove - $ (n!)^2 ≤ n^n(n)!$ For $ n = 1 $, LHS $≤$ ...
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2answers
109 views

$34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$

There was a number theory question that I have to do for homework. $34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$ I know $b=0$ because $10^7\big|34!$ only. But how can ...
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1answer
62 views

Prove that $n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$ [closed]

Prove that if $s_n=1^1+2^2+3^3+\cdots+n^n$ then $$n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$$ (it holds for $n$ larger than $2$). I want to prove ...
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1answer
104 views

Four elevators stop on floors $1$ to $8$; a separate elevator stops on $1$ and $4$. Suppose you want to get from $1$ to $4$ …

I recently encountered an interesting math problem at the hospital. The hospital had two elevator banks. The first one had four elevators that went to any of the floors, 1 through 8. While the second ...
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2answers
107 views

How does $a^{a^n}$ compare with $n!$ asymptotically?

I am learning Asymptotic complexity of functions from CLRS. I know that exponentiation functions like $a^n$,$(a>0)$ are faster than $n!$ But what about $a^{a^n}$ vs $n!$ How do they compare? A ...
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1answer
54 views

Distribution of digits across all factorials

Show that the distribution of zeroes across all digits of all n! for $n\in \mathbb{N}$ converges to $ \frac{1}{6} $ and hence, $\frac{5}{54}$ for all other digits 1 through 9. In other words, let's ...
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1answer
41 views

How do we obtain the following: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$

I saw the following equality in an informal proof: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$ I did not understand it and ...
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3answers
53 views

$100! $ in terms of $2^m Z$

Question I have encountered an question. If $$ 100! = 2^m Z $$ Where $Z\notin2\mathbb Z$ is an integer, find $m$ where $m \in \mathbb{ Z^+} $ My Attempt As $$100! = 2^{50} 50!$$ $[ 1×3×5×6. . . × ...
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2answers
143 views

Find all pairs $(k, n)$ of positive integers such that $k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$

Find all pairs $(k, n)$ of positive integers such that $$k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$$ I tried to solve this problem but only found one solution $(1,1)$. Please help me to ...
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2answers
37 views

Permutations after shuffling and drawing multiple cards from a deck

I'm trying to calculate the number of possible configurations of "starting game states" for a game that starts with shuffling a deck of 60 cards and drawing 7 cards. Some of the cards in the 60-card ...
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2answers
90 views

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct?

Can $n$ be expressed as $a^b-c^d$, where $a,b,c,d,$ and $n$ are natural numbers, not necessary distinct, and $b$ and $d$ can not be both equal to $1$? For example, when $n=1319$, then $2^{11}-3^6=...
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1answer
76 views

Prove that $~~~~~a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+…+a_m\right)! ~~~~~~~~~\forall ~~~a_1,a_2,…,a_m\in N$

Show that $$a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+...+a_m\right)! \forall a_1,a_2,...,a_m\in N$$ Case 1 $m=2$. We need to prove $$a_1!a_2!\mid (a_1+a_2)!$$ $a_1+a_2=1$ and $a_1+a_2=2$. It is ...