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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0
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1answer
26 views

Finding maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
3
votes
1answer
59 views

Exponent analog to the factorial function

Triangular numbers can be discovered by taking any number $n$, and adding $$\sum_{i=0}^n i = n + (n - 1) + (n - 2) ... 1 = \frac{n(n + 1)}{2}$$ These numbers can be generalized by putting any real ...
17
votes
5answers
870 views

A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
4
votes
3answers
2k views

Notation for factorial-type pattern with a skip/step of two instead of one?

I came across a peculiar pattern when solving a recurrence relation today: Some sequence $a_n$ looks as such: $a_0 = 1$ $a_2 = \frac{1}{2 \cdot 1}$ $a_4 = \frac{1}{4 \cdot 2 \cdot 1}$ $a_6 = ...
0
votes
0answers
30 views

Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
1
vote
1answer
721 views

Number of Divisors of N factorial

Say $d(N) =$ Number of factors of $N!$ Briefly: I wish to know if there is a Recurrence relation for this problem. Now I wish to Know if there is a way to calculate $d(N)$ in terms of previously ...
-3
votes
0answers
47 views

Evaulate the expression [closed]

$$x^2 + \frac53x^3 + \frac{23}{12}x^4 + \frac{119}{60}x^5 + \dots + 2\times \frac{5000!-1}{5000!}x^{5000}$$ How to evaluate the expression for $x=0.7893$ to the nearest $20$ decimal places?
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0answers
24 views

multifactorial of non-integer

I want to calculate 12.1!!!!!! , Just for curiosity. (One of my friend texted the term to me for some complex reason.) I searched for multifactorial in terms of gamma function or equation, and found ...
3
votes
1answer
155 views

Uniquely identify any finite subset of an infinite set

Let $U$ be an unbounded subset of $\mathbb{N}$. Let $D = \mathcal{P}_{<\omega}(U)$ (the set of all finite subsets of $U$). Let $f$ be an injection such that: $f: D \rightarrow \mathbb{N} $ ...
0
votes
3answers
118 views

determining the greatest $n$ for which $3^n$ divides $30!$

Determine the greatest integer $n$ such that $3^n\mid 30!$ I have no idea of how to approach this problem. I would first calculate $30!$ but obviously that number is way too large. Any help?
3
votes
0answers
105 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
9
votes
7answers
360 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...
6
votes
4answers
9k views

The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)

How can we prove, without using the properties of binomial coefficients, the product of n consecutive integers is divisible by n factorial?
0
votes
0answers
19 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
1
vote
2answers
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)! ...
0
votes
2answers
35 views

Mathematical Induction with series and factorials.

I wish to show the following $$ a_{n}=\sum_{k=0}^{n}\frac{1}{(2k+1)!(2(n-k)+1)!}=\sum_{k=0}^{n+1}\frac{1}{(2k)!(2(n+1-k))!}=b_{n+1} $$ for $n\geq0$ and wish to do it using induction. I've shown it ...
1
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3answers
703 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
2
votes
2answers
48 views

Factoring the factorials

Just for the fun of it, I've started factoring $n!$ into its prime divisors, and this is what I got for $2\leq n\leq20$: $$\begin{align} 2! &= 2^\color{red}{1} &S_e=1\\ 3! &= ...
2
votes
2answers
61 views

Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By ...
2
votes
3answers
155 views

Can non-integer factorials be calculated without numerical integration?

I saw a strange way to write the factorial function somewhere and after some integration by parts, it all sure enough worked out. $$ n! = \int_0^\infty x^{n}e^{-x}dx $$ $$ ...
-1
votes
3answers
52 views

Proving that $\frac{(N+p-1)!}{p!(p-1)!N!}$ is an integer?

Consider the quantity: $$\frac{(N+p-1)!}{p!(p-1)!N!}$$ where $N$ and $p$ are positive integers. How can we show that this is always an integer (which I believe has to be the case since it represents ...
2
votes
1answer
75 views

Is there an equivalent to the Bertrand's postulate between factorials and primorials?

As the title explains, I am trying to know if there is a definition about the upper limit to find the first primorial $p_i\#$ (following the definition at OEIS) existing after a given factorial $n!$ ...
1
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1answer
280 views

Two Questions about Gamma Function Terminology

Gamma function is also known as generalized factorial function . 1. Why does the term "generalized" have been used? 2. Why is the Gamma function called Euler's second integral?
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2answers
47 views

Suppose a coin in tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences…

Suppose a coin is tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences are there in which there are at least $5$ tails in a row? I know this is Permutation with repetition. My ...
1
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4answers
28 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus ...
0
votes
1answer
34 views

Trying to Express A Factorial As A Polynomial

I'd like to express the following as a polynomial. $$(a-1)(a-2)(a-3) . . . (a-b)$$ where $b<a$ I'm currently working on it now, but wanted to see if anyone's already done it, or already know what ...
1
vote
3answers
47 views

Demonstrating that 1! is = 1

The problem with this explanation is that it's using n = 2 instead of n = 1. Please read the explanation I found on "Math Forum - Ask Dr. Math" ( http://mathforum.org/library/drmath/view/57128.html ). ...
2
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2answers
63 views

solve for $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$

I believe the $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$ -> 0. But I am not sure if my reasoning is correct. Because there is a higher power in the denomination that the numerator, the ...
2
votes
2answers
38 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
3
votes
0answers
39 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
3
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0answers
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prove that $(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$ [duplicate]

prove that $$(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$$ I tried to prove by the induction that $(\frac{n}{3})^n<n!$ and $n!<e\cdot(\frac{n}{2})^n$, but I failed my assumption ...
0
votes
1answer
99 views

Proving an inequality involving factorials: $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$ [duplicate]

For $n \geq 6$, where $n$ is a natural number, prove that $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$. I tried using induction but could not do it.
14
votes
1answer
2k views

Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$

I met an inequality, I ask, do not mathematical induction to prove that: Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction
2
votes
0answers
55 views

Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.

Find the maximum and minimum value of the function $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}.$ ATTEMPT:- By A.M.-G.M. inequality, $\frac{a+b}{2}\ge\sqrt{ab}$, $\quad$ for $a,b\gt 0$ with equality ...
0
votes
1answer
43 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
-12
votes
3answers
366 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 ...
4
votes
2answers
23k 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
99 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?
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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 ...
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0answers
22 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
70 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
68 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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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
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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
209 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
79 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?
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
55 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 ...