Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms (1)

0
votes
1answer
34 views

solve for $\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$

I am having trouble solving for this limit with factorials. $$\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$$ Any hints or suggestions would be great
0
votes
0answers
47 views

in spanish do you have to do the upside down exclamation marks before a number for factorials? [on hold]

for example $4! = 4 \times 3 \times 2 \times 1 = 24$ if you were to write this in spanish would you have to write $¡4! = 24$ ? a quick google search doesn't give me anything
-9
votes
3answers
341 views

Is it possible to calculate $\int x! dx$ [on hold]

Is it possible to calculate $\int x! dx$, if yes ,then how and if no ,then why not? This question came in my mind when, I solved some questions on integration. Until now I haven't got the right ...
4
votes
2answers
22k views

What does ! mean in sequences?

I'm doing a sequences problem where I have to write the first five terms of a sequence. It looks normal, but there is an exclamation mark on the denominator: $$a_n = \frac{1}{(n + 1)!}$$ & ...
0
votes
1answer
98 views

finding the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$ [on hold]

What is the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$? How will I solve this type of problems?
1
vote
7answers
3k views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
0
votes
0answers
21 views

Proving that ${p \choose r}$ is an integer for a prime $p$ and $0 < r < p, r \in \mathbb{Z}$ [duplicate]

I need to prove that given integers $p$ and $r$ such that $p$ is prime and $0 < r < p$, ${p \choose r} = \frac{p!}{r!(p-r)!} \in \mathbb{Z}$ As of now, I don't have any ideas on how to proceed. ...
5
votes
1answer
68 views

The number of zeros in the expansion of $n!$ in base $12$

During an interview last year I was asked the following question: How many zeros appear at the end of $n!$ in base $12$, where $n$ is a positive integer? I applied the known Legendre formula for ...
5
votes
1answer
66 views

Is it true for $n > 2$ then there always exists a prime $\le n$ that does not divide $n$?

I was thinking of how to prove $\frac{n^n}{n!}$ is never an integer for $n > 2$. I think if I prove the above question, then this follows immediately.
0
votes
2answers
1k views

Exotic 6-horse race betting probabilities

I'm gearing up for horse racing season, and I'm trying to teach some fellow engineering friends how to bet "exotic" bets by using colored dice to simulate horses. So, the odds for each horse winning ...
1
vote
5answers
55 views

Prove $\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$ [duplicate]

Prove $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ Proof by induction: true for $n=2$. Assume true for $n$ and see if $n+1$ is true. $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ ...
5
votes
3answers
207 views

If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$

If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$ Well I was able to prove this intuitively, but what i need is a rigorous mathematical proof. I shall explain my ...
0
votes
3answers
76 views

Does (9/2)! have a real answer or not? [duplicate]

The TI-84 says 52.342777 but other calculators says domain error.
0
votes
0answers
25 views

$n!>a$, can we solve for $n$ in terms of $a$? [duplicate]

Can we explicitly solve $n$ in terms of $a$? Can we rewrite this inequality in the form of $n>f(a)$ without using $n!$ the factorial?
0
votes
1answer
34 views

Big O for factorials

Hello I have trouble proving:$$(n+1)!\notin O(n!)$$ My first step is the following: $$(n+1)!-cn!\le0$$ Can you please help me with the next step?
1
vote
3answers
52 views

What does an exclamation point raised to a power, with no preceding number, mean?

In the OEIS sequence A049210, I noticed an odd notation I haven't seen before: a(n) = (8*n-1)(!^8), n >= 1, a(0) = 1. What does the ...
16
votes
4answers
5k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
0
votes
1answer
61 views

Prove that $n! = O(n^n)$

I thought $n^n$ was greater than $n!$. How would I go about proving this? I have this so far: Assume that $P$($n$) is true $n!$ = O($n^n$) Assume that $P$($n+1$) is also true $(n+1)! ...
1
vote
1answer
37 views

The exponent of $11$ in the prime factorization of $ 300!$ is___.

The exponent of $11$ in the prime factorization of $ 300!$ is $27$ $28$ $29$ $30$ My attempt: According to Exponent of $p$ in the prime factorization of $n!$ ...
5
votes
4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
2
votes
5answers
397 views

Divisibility via Induction: $8^n\mid (4n)!$ [duplicate]

I have to prove by induction the following fact: Show that $(4n)!$ is divisible by $8^n$. My stab at the solution: I have a slightly bad feeling about this solution and I would like if someone ...
1
vote
2answers
31 views

Combinations equation solving with factorial

I was trying to solve the equation using factorial as shown below but now I'm stuck at this level and need help. $$C(n,3) = 2*C(n,2)$$ $$\frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$$ $$3! (n - 3)! = ...
1
vote
1answer
1k views

How to calculate the number of permutations and combinations if k is equal to n?

Say the question is How many unique ways are there to arrange the letters in the word FANCY? The formula I use for permutations is n! / (n - k)! ...
1
vote
1answer
37 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}$$. I am unable to do this one. ...
0
votes
0answers
28 views

Modifying permutation function for inputs with equivalent ratios

I have the following function: $$f(a_1,a_2,\ldots,a_n) = \frac{(a_1 + a_2 + \cdots +a_n)!}{(a_1! a_2! \cdots a_n!)}$$ where $a_i\ge 0$ I need to modify this function such that $f(a_1,a_2,...,a_n) = ...
1
vote
3answers
67 views

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$ So far I have: Base case (n = 1) = $8^{1} | (4(1))!$ = $8 | 24$ which is true. Induction Step: $8^{n + 1} | (4(n + ...
1
vote
2answers
35 views

Congruence Modulo involving factorials

How do I show that $23!\equiv 21! \pmod{101}$? I tried using a calculator but the numbers are so big that am finding it hard to prove. How can factorials be broken down so that they can be easily ...
0
votes
1answer
44 views

Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
12
votes
4answers
184 views

Prove that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$ [closed]

Using a graphing calculator, it seems that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$. How can this be proven?
2
votes
2answers
52 views

How to solve equation with factorial using algebra?

I bring this sample in order to ilustrate $$x! = 2^x + 8$$ I know the answer is $x=4$ but I dunno how to prove it. I mean, if i put the number 4 by observation, tryal and error, I can get the ...
0
votes
3answers
53 views

Proof by mathematical induction that $2^n < (n+2)!$ for all $n\ge 0$

I have been trying to get this.. For hours. Prove by M.I. that $2^n < (n+2)!$ for $n\ge0$ Here is what I am doing: Base case checks out at $n=0$ Make assumption for: $n=k$ Want to prove: ...
2
votes
2answers
41 views

How to show using proof by induction: $\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$

I'm having quite a few problems with the following proof by induction question: $$\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$$ I manage to do the easy parts of the base step ($n=1$) ...
0
votes
1answer
45 views

Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: ...
1
vote
1answer
33 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
2
votes
1answer
112 views

how is a factorial fraction equal to the product notation

How is the $\prod_{k=2}^n(2k-3)={(2n-3)!\over 2^{n-2}(n-2)!}$, where $n \geq 2$ Note: I know that the $(2n-3)!$ is equal to the product of $2k-3$ from $k=2$ to $n$, but I can't figure out the bottom ...
2
votes
4answers
7k views
2
votes
1answer
273 views

Does the series converge or diverge? $\sum_{n=1}^\infty\frac{4^n+n}{n!}$

Does the following series converge or diverge? $$\sum_{n=1}^\infty\frac{4^n+n}{n!}$$
6
votes
3answers
92 views

Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that$\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
12
votes
3answers
140 views

Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\displaystyle \sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}$ for $1 \leq \alpha \leq p$? EDIT: This is a result that I derived ...
6
votes
1answer
80 views

Calculating $\sum_{k=1}^nk(k!)$ combinatorially

The sum $\sum_{k=1}^nk(k!)$ can be easily calculated by noting $k(k!)=(k+1)!-k!$. Is there a way to calculate the sum nicely using a combinatorial argument. Is it possible to notice it is $(n+1)!-1$ ...
6
votes
1answer
57 views

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$ [duplicate]

Combinatorial argument for $$1+\sum\limits_{r=1}^{r=n} \ r\cdot r! = (n+1)!$$ The algebraic proof is easy as $r=(r+1)-1$.
-1
votes
1answer
17 views

Fractional numbers' factorial [duplicate]

Is there a law or anyway to know the factorial of a fractional number, because as I see the law of factorization n! = n x (n-1) x (n-2) x ... x 3 x 2 x 1 isn't ...
0
votes
1answer
19 views

function representation of power series

What is the function representation of this power series? [Summation from n=0 to infinity of ($x^n)(n+1)!/n!$ The solution is $\frac{1}{(1-x)^-2}$ but how??? I know that ...
7
votes
2answers
158 views

Is there a name for an 'incomplete' factorial $\frac{n!}{m!}$?

I noticed I was computing $${n! \over m!} ,$$ where $n > m$, inefficiently, as $$\frac{\prod_{k=1}^{n} k}{\prod_{k=1}^m k},$$ when many terms cancel out and I could just be calculating ...
1
vote
2answers
52 views

Problem involving factorials (divisibility) [closed]

Show that, for every $n \in \Bbb N$, the following number is natural: $$\frac {(n!)!} {{n!}^{(n-1)!}}$$. I dont't know how to prove, as I tried to find a way including combinatorics.
1
vote
2answers
105 views

Calculate sum of infinite series by solving a differential equation

Calculate the sum of the infinite series $$\sum_{n=0}^{\infty}\frac{1}{(3n)!}$$ by solving an aptly chosen differential equation. I know that one can solve a differential equation by assuming that ...
6
votes
5answers
29k views

Prove by induction that $n!>2^n$ [duplicate]

Possible Duplicate: Proof the inequality $n! \geq 2^n$ by induction Prove by induction that $n!>2^n$ for all integers $n\ge4$. I know that I have to start from the basic step, which is ...
6
votes
2answers
4k views

Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
6
votes
1answer
201 views

Factorials and the Mod function

I was just playing with the factorial and the modulo function. I just observed this interesting property. I was using a calculator $$13!\equiv 13\times 12\pmod{169}\\ 17!\equiv 17\times ...