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

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1answer
100 views

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

What is the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$? How will I solve this type of problems?
5
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1answer
71 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
72 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.
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5answers
60 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
212 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
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3answers
82 views

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

The TI-84 says 52.342777 but other calculators says domain error.
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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?
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1answer
35 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?
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3answers
57 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 ...
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3answers
392 views

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

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 ...
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1answer
42 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 $n!$ $\left\lfloor\dfrac{300}{11}\...
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2answers
32 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
39 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
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0answers
29 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) = ...
2
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5answers
403 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 ...
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3answers
71 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))...
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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
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1answer
63 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)! ...
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3answers
58 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^...
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2answers
44 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
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1answer
49 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: $$v_p(m!)\leqslant\frac{m}{p-1}....
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1answer
46 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 ...
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1answer
36 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: $a=2$, ...
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1answer
20 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 ...
12
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3answers
142 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 ...
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2answers
55 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 ...
6
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1answer
60 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 $r=(r+1)-1$.
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1answer
20 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 $\sum_{n=0}^{\infty}(x^n)/n!...
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2answers
55 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.
6
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1answer
205 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 $$13!\equiv 13\times 12\pmod{169}\\ 17!\equiv 17\times 16\pmod{289}$$...
0
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1answer
32 views

Trying to simplify an expression for an induction proof.

I got it down to $(k+2)!-1 + (k+1)((k+1)!)$ I am trying to get it to $(k+2)!-1$ but I guess I do not understand factorials enough to simplify this. I am also assuming I am doing the induction ...
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2answers
115 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 ...
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0answers
35 views

Using mathematical induction to prove P(n) [duplicate]

I have the statement $P(n)$: $2^n<(n+1)!$, for $n \geq 2$; $P(2)$: $2^2 < 3!$ which is true I.H P(k): $2^k<(k+1)!$ show that $P(k+1)$: $2^{k+1} <(k+2)!$ Here is my approach: $2^k<(...
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2answers
59 views

Infinite Sum Calculation

How do we calculate this? $$ \sum_{n=1}^\infty \frac{n-1}{n!}$$ From my readings on previous posts here and through Google, I found that: $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ But then how do I ...
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3answers
208 views

Divergent sum of factorials

Is it possible to get an exact value of the sum (using divergent series summation methods) $$ \sum_{n=0}^\infty~ \frac{(n+k)!}{n!} \quad?$$ where $k$ is a positive integer. The only other divergent ...
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1answer
18 views

Simplifying an equation involving factorials.

I have found that the residue of the function $f(x) = \frac{1}{(1+x)^{n+1}}$ is $R= 2\pi i\frac{(n+1) \cdot (n+2) \cdot ... \cdot (2n)}{n! (2i)^{2n+1}}$. I am having trouble with showing that ...
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0answers
28 views

Divison in factoradic base

I'm trying to find am means of dividing two numbers in factoradic base. So far goolging seems to turn up nothing at all. Is there a better way of doing this than long-division? I'm hoping for ...
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1answer
38 views

How does factorials work in algebraic equations?

I was reading a text book and found these two lines and I have no clue how did step1 become step2. Please help me with this. Thanks Step 1: $$ \frac {5!}{(4-r)!} = \frac {6*5!}{(5-r+1)(5-r)(5-r-1)!} ...
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1answer
28 views

Evaluation of an expression

I have difficulties to evaluate this expression to the desired result. (It is a proof based on mathematical induction, left = right) $(k+1)!-1+(k+1)*(k+1)! = (k+2)!-1$
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0answers
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What is $\prod_{k=1}^{(p-1)/2} (2k-1)^2 \bmod p$ where $p$ is an odd prime?

What is $\prod_{k=1}^{(p-1)/2} (2k-1)^2 \bmod p$ where $p$ is an odd prime? Note: Someone just asked this and it was deleted while I was working on it, so I am posting it with my answer. I get $(-1)^...
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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 $$\...
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1answer
32 views

What function describes this problem of every possible breeding of a set of dogs?

If I have n dogs [a, b, c, ...], and I want to breed them in every possible combination (every possible binary tree made of ...
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5answers
94 views

How one can show $(r!)^s$ divides $(rs)!$?

I would like if anybody have suggestions to prove: if $n = rs$ with: $r > 0$ and $s > 0$ then $(r!)^s \mid n!$ suggestions?
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2answers
167 views

Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of k and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum ...
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2answers
55 views

Integral To Summation Problem

$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n (-1)^i\,\frac{...
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1answer
49 views

Proof By Induction With Integration Problem

I am required to prove this formula by induction$$ \int x^k e^{\lambda x} = \frac{(-1)^{k+1}k!}{\lambda^{k+1}} + \sum_{i=0}^k \frac{(-1)^i k^\underline{i}}{\lambda^{i+1}}x^{k-i}e^{\lambda x}$$ where $...
1
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1answer
186 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor x\rfloor-n}}=\frac{\Gamma{(x+1)}}{\Gamma{(x-\...
2
votes
3answers
60 views

Is $n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)^i}{i!}}+…+\sum_{i=0}^{2}{\frac{(-1)^i}{i!}}\bigg]=(n-1)!$ true?

I am in the middle of doing a problem and has this sort of expression. I have a feeling that the following equality holds: $$n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)...
0
votes
1answer
31 views

Negative factorial over another negative factorial [closed]

Suppose that you wanted to take the ratio of two negative factorials, such as: $\frac{(-n)!}{(-m)!}$ for positive integers $n,m>0$. I am pretty sure it is well defined (binomials with a negative ...
0
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1answer
94 views

Finding the $n^{th}$ derivative of $x^r$

I'm looking for a non-piecewise function -- $g(n,x)$ -- that satisfies this equation: $g(n,x)=\large\frac{d^{n}}{dx^{n}}x^{r}$ Where $n\in\Bbb{Z}$ and is the $n^{th}$ derivitive of $f(x)$ I ...