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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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0answers
51 views

Maths based factorial function puzzle [on hold]

This is a geocache puzzle and the GC code of the puzzle is GC7VWAA The puzzle is correct in the image above. $$\begin{array}{lcr} (G-C)!+1&\leftrightarrow&6046750\\ \bigl((G-C)^2+\sqrt{\...
2
votes
1answer
69 views

$(\frac{n}{e})^{n} < n! < (\frac{n}{e} + n\varepsilon)^{n}$ doesn't comply with the limit definition?

I try to understand what I've overlooked, when I came up with this inequality: First, we have this limit: $$\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} = \frac{1}{e}$$ Which gives, by the ...
3
votes
3answers
68 views

Longest sequence of consecutive integers which are not coprime with $n!$

For any integer $n$, the factorial $n!$ is the product of all positive integers up to and including $n$. Then in the sequence $$n!+2,n!+3,... ,n!+n$$ the first term is divisible by $2$, the second ...
1
vote
2answers
47 views

Check if $k$ is divisible by $2^9$ or $2^{10}$

$k=\frac{512!}{256!*128!*...*2!*1! } $ I need to check if the expression k is divisible by $2^9$ or $2^{10}$. This is a multiple choice question and the options and the question goes like this: ...
1
vote
2answers
36 views

Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N??

Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N? where a is an arbitrary integer, and N is a prime. e.g. i know 22! mod 23 ≡ 22 using Wilson's theorem. lets say i want to know ...
-4
votes
1answer
44 views

Factorial problem [closed]

I am really sorry for not inputing it manually, I am just really in a hurry, apologies.
4
votes
3answers
66 views

Can't find a seemingly simple limit $\lim_{n\to\infty}\frac{(n+k)!}{n^n}$

Evaluate the limit: $$ \lim_{n\to\infty}\frac{(n+k)!}{n^n}, \ n,k\in\Bbb N $$ I would like to avoid Stirling's approximation, derivatives and Cesaro-Stolz, since none of them has been yet ...
0
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0answers
27 views

A question on the approximation of $\ln(x!)$ for small x

In a question of Arfken and Weber's Mathematical Methods for Physicists, For small values of $x$, $\ln(x!)=-\gamma x+\sum\limits_{n=2}^{\infty}(-1)^n\frac{\zeta(n)}{n}x^n$, where the symbols used ...
0
votes
2answers
66 views

Show that $\frac{(2n)!!}{(2n+1)!!}$ converges.

Given a sequence $\{x_n\}, \ n\in\Bbb N$: $$ x_n = \frac{(2n)!!}{(2n+1)!!} $$ Show that $x_n$ converges. I'm wondering why I'm getting a seemingly wrong result (assuming the problem statement ...
6
votes
6answers
119 views

Determining the missing digits of $15! \equiv 1\square0767436\square000$ without actually calculating the factorial

$$15! \equiv 1\cdot 2\cdot 3\cdot\,\cdots\,\cdot 15 \equiv 1\square0767436\square000$$ Using a calculator, I know that the answer is $3$ and $8$, but I know that the answer can be calculated by ...
3
votes
0answers
93 views

Methods for proving a function outputs an infinite number of integers

I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in ...
-1
votes
4answers
76 views

Sum of $\sum_{n=1}^\infty \frac{(2n-1)!}{(2n+2)!}$ [closed]

I would like to have some help finding the sum of $$\sum_{n=1}^\infty \frac{(2n-1)!}{(2n+2)!}$$ I am trying to solve it for $2$ days already, and it's getting harder and harder everytime!
1
vote
2answers
75 views

Probability an ace lies behind first ace

Consider a deck of 52 cards. I keep drawing until the first ace appears. I wish to find the probability that the card after is an ace. Now, the method I know leads to the correct answer is that given ...
0
votes
1answer
79 views

Calculate $\sum_{k=l}^{2n} \binom{2n+k}{2k} \frac{(2k-1)!!}{(k-l)!} (-1)^k$ [closed]

I'm trying to proof that $$\sum_{k=l}^{2n} \binom{2n+k}{2k} \frac{(2k-1)!!}{(k-l)!} (-1)^k = \begin{cases} 0 \quad {\rm if} \, \, l \, \,{\rm odd} \\ \frac{(-1)^{n-l/2}(2n+l)!}{4^n \left(n-\frac{l}{2}\...
0
votes
1answer
33 views

Need help understanding a step in an induction proof

I want to prove that $(n!)^{(n-1)!}$ divides $ n!!$ via induction. I was going through a post I found on Quora doing this, but I got hung up on the last step. For the sake of legibility I'll rewrite ...
1
vote
2answers
43 views

Further explanation for steps of an equation that proofs that $\sum^{n}_{k=0}k\cdot \binom{n}{k}=n\cdot2^{n-1}$

So, this is one of the questions in my textbook, which seems to be quite common: $$\sum^{n}_{k=0}k\cdot \binom{n}{k}=n\cdot2^{n-1}$$ The same book provides the following solution: $$\sum_{k=0}^{n}k\...
1
vote
2answers
40 views

why is $\frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$?

This is an equation from my textbook that I am trying to understand: $$ \frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$$ What I got so far, is that $\frac{k\cdot n!}{k!} = \frac{...
1
vote
2answers
48 views

Turning product sequences into factorials

I am trying to figure out the steps between these equal expressions in order to get a more general understanding of product sequences: $$\prod_{k=0}^{n}\left(3n-k\right) + \prod_{k=n}^{2n-3}\left(2n-k\...
0
votes
2answers
36 views

Simplyfying factorials why is $(n+1)(n-1)!-(n-1)! = n(n-1)!$?

I am a bit unclear on how these two expressions are equal: $$(n+1)(n-1)!-(n-1)! = n(n-1)!$$ So far, I obtained $$\frac{(n+1)!}{n}-(n-1)!=\frac{(n+1)n(n-1)!}{n}-(n-1)! = (n+1)(n-1)!-(n-1)!$$ ...
0
votes
2answers
47 views

What would be the sequence of $n!!$ (double factorial)

I know that the sequence of $n!$ is $$n(n-1)(n-2)\cdots(2)(1)$$ but what would be the sequence of $n!!$? (In the interest of clarity, this is also known as the double factorial, not to be confused ...
1
vote
3answers
66 views

How find the limit of $\lim\limits_{n\to \infty }\left(n-1\right)!$

I have the next limit: $$\lim\limits_{n\to \infty }\left(\sqrt[n]{\left(\frac{n!-1}{n!+1}\right)^{\left(n+1\right)!}}\right)$$ I had done some steps and simplified it to: $$\lim\limits_{n\to \infty ...
7
votes
3answers
159 views

Is it possibile to obtain the sum of the following series without using hypergeometric functions?

I know that the following series: $$ \sum_{n=1}^{+\infty}\frac{ (n!)^2}{(2n)!} $$ converges. If I plug it in Wolphram Alpha, I can see that its sum is $$ \frac{1}{27} \left(9 + 2 \sqrt{3} \pi\right). $...
0
votes
3answers
40 views

Prove $2\times1!+5\times2!+10\times3!+…+(n^2+1)n!=n(n+1)!$ for all positive integers

I am trying to prove by mathematical induction $2\times1!+5\times2!+10\times3!+...+(n^2+1)n!=n(n+1)!$ for all positive integers $n$. So far I have: Solved in the first case possible - $1$ Assumed ...
2
votes
1answer
45 views

Finding Bounds in proof of Stirling's Formula

In a proof of Stirling's Formula, my professor claims that $\frac 12 $($ln$k + $ln$(k+1)) $\le$ $\int_k^{k+1}$$ln$x $dx$ $\le$ $\frac 12$($ln$k + $ln$(k+1)) + $\frac 1{k^2}$. I can see that the ...
2
votes
3answers
47 views

Number of divisors of $10!$

Determine the amount of divisors of $10!$ This is a question in my combinatorics textbook, so I need to somehow reduce this to an elementary counting problem like combinations, permutations with or ...
1
vote
5answers
103 views

Show that $\lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty$

Show that $$ \lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty $$ The only way i've been able to show that is using Stirling's approximation: $$ n! \sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n $$ Let: $...
0
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0answers
26 views

How to get this closed form for such recurrence?

We have for $k>0$, $n>0$, $m\geqslant0$ $$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$ also $$p_0(n,m)=\begin{cases} (n-1)!,&\text{$n>0, m=0$}\\ 0,&...
0
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0answers
29 views

Intuitive method for solving $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$

Is there any intuitive method of solving: $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$ without having to develop the Exponential Integral Ei?
8
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2answers
145 views

Geometrical interpretation for the sum of factorial numbers

I am in need of a way to represent the sum $1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$ in a geometrical way. What I mean by this is that for example, the sum $1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 =...
4
votes
5answers
137 views

limit of $\left(1+\frac{1}{n!}\right)^n$

I'm having trouble resolving the following limit: $$ \lim_{n \to \infty} \left(1+\frac{1}{n!}\right)^n $$ Intuituvely the limit is equal to 1, but the exercises requires me to resolve via calculation ...
0
votes
0answers
38 views

Product of evenly spaced factorials

I am attempting to get a closed form expression for a product of factorials $$(a!)\cdot(2a)!\cdots (na)!$$ where $a$ is a positive integer. Mathematica seems able to compute this whichever $a$ I give ...
0
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2answers
53 views

Spm test question(Combination) [closed]

$\binom{y}{m}=\binom{y}{n}$, How should I express y in terms of m and n?
3
votes
4answers
58 views

Factorial Proof - ${n \choose r-1}+{n \choose r}={n+1 \choose r}$

${n \choose r-1}+{n \choose r}={n+1 \choose r}.$ So what I tried to do was expand the first and second term. $\frac{n!}{(r-1)!(n+1-r)!}+\frac{n!}{(r)!(n-r)!}.$ Then what I did was try to get common ...
0
votes
1answer
48 views

Permutations vs. Combinatorial vs. Factorials vs. Exponents

I'm currently working on a probability course, and I am constantly having trouble figuring out when to use permutations vs. combinations vs. factorials vs. exponents in order to calculate sample size, ...
2
votes
1answer
86 views

Factorial like product

I am trying to solve http://www.javaist.com/rosecode/problem-519-Factorial-like-Product-askyear-2018 Restating the problem: $$R(p,k)=\prod_{i=1}^{p-1}(i^k+1) \hspace{2 mm} \text{mod}\hspace{2 mm} p$$ ...
0
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2answers
67 views

What is the best estimation of the factorial function?

I have to calculate factorials of an arbitrarily large integers mod another arbitrarily large integer. i have considered Stirling's Approximation and Ramanujan’s factorial approximation. is it ...
2
votes
2answers
87 views

How to solve the limit $\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$.

How to solve this limit?? $$\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$$ It's a limit, not a series
0
votes
0answers
32 views

Trying to solve “(2n+3)!/(2n+1)!” algebraically and my calculator gives different results

$$\frac{(2n+3)!}{(2n+1)!}$$ I solved the above problem and I got $(2n+3)(2n+2)$, but I inputted it onto my TI92 plus. It gives me: $\frac{(2n+3)!}{(2n+1)(2n)!}$ Any ideas?
0
votes
0answers
77 views

Does $\aleph_0!=\omega$? [duplicate]

More generally, is the order type of some cardinal $\alpha$ equal to $\alpha!$? Related What is $\aleph_0!$? Factorial of Infinite Cardinal factorial of infinite Cardinals
2
votes
0answers
54 views

Results from extensions and variations of Dixon's Identity

The following problem\s stem from my investigations regarding an obscure identity discovered by Sir Alfred Cardew Dixon of Cambridge, it can be considered a theorem or formula, it's just silly ...
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1answer
28 views

Multiplication of two factorials

Please guide me whether the two sides are equal: $$\Big(\frac{(2-a)b}{c}\Big)!*\Big(\frac{1}{c}\Big)!=\Big(\frac{(2-a)b+1}{c}\Big)!$$ where a, b and c are constants >0, Kindly prove this equation.
0
votes
0answers
20 views

Closed form for $q_k(0,0)$ from recurrence

We have $$p_0(n,m)=\begin{cases} 0,&\text{$n=m=0$}\\ (n-1)!,&\text{$n>0, m=0$}\\ 0,&\text{$n\geqslant0, m>0$} \end{cases}$$ $$q_0(n,m)=0, n\geqslant0, m\geqslant0$$ and $$p_k(n,m)=\...
4
votes
1answer
78 views

Help understanding the cause of this pattern when writing π as an infinite series with double factorials

I made a post about a year and a half ago: $\pi$ as an Infinite Series using Taylor Expansion on Equation of a Circle where essentially I used the Taylor series expansion on $\ y = \sqrt{r^2-x^2}$ (...
4
votes
3answers
72 views

Prove that $\sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}=\frac{n!}{x(x+1)\cdots(x+n)}$.

Given the following formula $$ \sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}\,. $$ How can I show that this is equal to $$ \frac{n!}{x(x+1)\cdots(x+n)}\,? $$
3
votes
4answers
58 views

All positive integers a and b such that $a! + 6 = b^2$

I'm not sure how to approach proving solutions for this problem. I wrote a python program which shows a must be $\geq 30$, but I don't understand why. ...
1
vote
0answers
24 views

How to solve this limit (with factorial)? [duplicate]

How to solve this limit?? $$\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$$ It's a limit, not a series
0
votes
0answers
25 views

Generalizing rising and falling factorial to complex arguments preserving zeroes

Falling factorials count injective functions. There are no injective functions $ A \to B $ if $|A| \gt |B|$ . Is there a way to extend the definition of the falling factorial, ideally to $\mathbb{C} \...
0
votes
2answers
82 views

How would I prove ${n^{2n} \gt (2n)!}$ using mathematical induction?

This is what I have done. I checked for $n=1,2$ and $3$ in the first step. I did the assumption for $n=k$ and the claim of $n=k+1$ in the second step, but I don't understand how the step no. $3$ works ...
1
vote
2answers
79 views

Inverse factorial function

I am wondering what is the inverse/opposite factorial function? e.g inverse-factorial(6)=3 Furthermore, I am intrigued to know the answer to: a!=π find a I would really appreciate if anyone could ...