Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
1answer
37 views

Factorial solve without substitution

I'm in stuck with this dimostration. I've got $$\frac{n!}{(n+1)!}$$ and it's must be $$\frac{1}{n+1}$$ If I put n=3, I've got $$\frac{3!}{(3+1)!}=\frac{1}{4}=\frac{1}{3+1}$$ and it's correct. But I ...
0
votes
1answer
12 views

Method of Differences/Partial fractions with factorials

By first expressing $\frac{1}{r!(r+2)}$ in the form $\frac{A}{(r+2)!} + \frac{B}{(r+1)!}$, find $\sum\limits_{r}^n \frac{1}{r!(r+2)}$. Struggling to do the partial fractions to begin.
0
votes
0answers
47 views

How many digits does $100!^{100!}$ have? [on hold]

How many digits does $100!^{100!}$ have? Can someone give the approximate value?
0
votes
3answers
93 views

Is $10^{100}$ (Googol) bigger than $100!$?

Is $10^{100}$ (Googol) bigger than $100!$? If $10^{100}$ is called as Googol, does $100!$ have any special name to be called, apart from being called as "100 factorial"? I ask this question ...
0
votes
3answers
51 views

Limit of $\lim\limits_{x \rightarrow 1} \frac{ (x +x^2+x^3+ \cdots +x^n)-n}{x-1}$ [on hold]

What is the limit of $$\lim\limits_{x \rightarrow 1} \frac{ (x +x^2+x^3+ \cdots +x^n)-n}{x-1}$$
0
votes
1answer
27 views

How do you solve the equation x!=n, for any value of n?

About 5 years ago, I did some research into factorials. I came across a problem online that asked me to solve x!=6. The answer was 3, of course, but when I tried to solve x!=3, I found no value of x ...
5
votes
1answer
93 views

How to solve $x^2=x!$

If $x^2=x!$ , what are the values of $x$? And this equation too, if $x!=120$, we know that here $x=5$ because $5!=5×4×3×2×1=120$. This seems something like back calculation. Is there any process to ...
0
votes
2answers
46 views

How to prove that $ n < n! - 1 $ for $n > 2.$? [on hold]

How to prove that $ n < n! - 1 $ for $n > 2.$? I have tried it by induction but I got stucked in the induction step in proving $ n +1< (n + 1)! - 1 $ for $n + 1> 2$. Could anyone help ...
1
vote
1answer
21 views

Proving a limit for factorials

On a site generalizing the factorial function, it states, "The probability of repetition gets negligible; we find that the ratio between x!/(x−r)! and x^r approaches 1 as x grows large while r is held ...
0
votes
1answer
28 views

How is it that (p-1)! is not congruent with 0 mod p if p is prime?

Why is this statement true? If $p$ is prime then $(p-1)! \not\equiv 0\space mod\space p$. I would like to know why this is true.
0
votes
1answer
29 views

Factorial$(kn)!$ expansion

I am try to expand the factorial $(kn)!$ And got this $$(kn)!=k^{kn}×n!×\prod_{i<k}{(n-\frac{i}{k})}$$ Is my approach right or contain any mistake. I calculated using induction Like n!, (2n)!, (3n)...
0
votes
3answers
37 views

$\prod$ and factorial

$\prod_{i=0}^{j-1}(j-i+1)$ is $=$ to $(j+1)!$ or $\le$? I think it is $=$, but if it's not, please put an explanation.
2
votes
1answer
41 views

Proof of inverse factorial decomposition $\frac{1}{n!}=\sum\limits^n_{i=1}\frac{(-1)^{n+i}}{(i-1)!(n+1-i)!}$

Doing research in probability modelling I obtained a decomposition of inverse factorial: $$\frac{1}{n!}=\sum\limits^n_{i=1}\frac{(-1)^{n+i}}{(i-1)!(n+1-i)!}$$ How can it be proven directly? In what ...
0
votes
0answers
28 views

Moment generating function (find the probability)

The moment generating function of a random variable $X$ is given by: $$M(t) = (1/3^{2k})(7+2e^t)^k, \quad \forall t$$ a) Determine $P(X = 3)$ b) Derive the $r^{th}$ factorial moment of $X$ I ...
0
votes
0answers
19 views

Summation simplification with factorial in denominator and combination in numerator

I want to simplify a below equation. $\sum^{K-1}_{v=0}{{{K-1}\choose{v}}\cdot\frac{x^v}{(v-1)!}}$ Is there any idea? Thanks in advance.
0
votes
1answer
45 views

Find the largest integer $k$ for which $3^k$ divides $400 \choose 200$.

I was working through a problem set for number theory and needed some help with this problem: Find the largest integer $k$ for which $3^k$ divides $400\choose 200$. I know this will reduce to $\...
2
votes
2answers
61 views

Double sum factorial manipulation

$$\sum_{B = 0}^{n-1} \sum_{A = 0}^{n-B-1} \frac{(n-1)!}{B!(n-B-1)!} \frac{(n-B-1)!}{A!(n-B-A-1)!} \frac{A}{A+B+1}$$ This is driving me nuts! Is there anyway to reduce $$\sum_{B = 0}^{n-1} \sum_{A = ...
0
votes
0answers
19 views

Minimum of maximum with factorial

Let $f:\mathbb N_+\to\mathbb N_+, f(n)=\min\{\max\{k!,(n-k)!,(n!-k!(n-k)!)\}|k\in\mathbb N_+, n-k> 0,\ n!-k!(n-k)!> 0 \}$. What's the asypmtotic growth of $f(n)$? Is it true that $f(n)=\Theta((...
0
votes
0answers
11 views

Has a Pattern been explored? The sequence of prime factors for a factorial expression?

I have been studying this sequence and I was wondering if there is any pattern to the sequence? Has this been explored? Is there a way to generate an expression that creates these sequences? Is it/...
0
votes
2answers
32 views

Laplace approximation of Poisson posterior from MacKay

I am doing exercise 27.1 on Laplace's method from David MacKay's textbook, which is to make a Laplace approximation of a Poisson model with an improper prior: $$ p(x \mid \lambda) = \frac{e^{-\lambda}...
3
votes
2answers
93 views

Factorials and prime numbers

Let $N$ be a positive integer not equal to $1$. Then note that none of the numbers $2, 3, \ldots, N$ is a divisor of $N! - 1$. From this we can conclude that: (A) $N! – 1$ is a prime number; (B) at ...
0
votes
1answer
20 views

Factorial representation with sum of products

I am a beginner in math I was developing a coding problem and here is the pattern which I found: Help me understand the pattern. Factorial of n will be the sum of all the product of all permutation ...
1
vote
1answer
33 views

Can someone explain how does step 1 render to step 2?

(1)$(n+1)!−1+(n+1)×(n+1)!$ (2)$=(1+n+1)×(n+1)!−1$ (3)$=(n+2)×(n+1)!−1 $ (4)$=(n+2)!−1$ I understand how step 4 derived from 3, but I am confused on how does step 2 derived from step 1? thank you
-2
votes
0answers
32 views

Calculation of the Digamma function

How might one prove that: $\psi\left(1\right) = -\gamma$ And how are its values calculated for non integer values?
1
vote
1answer
30 views

Diophantine with primes factorials

Let $p\in \Bbb{P},\alpha\in \Bbb{N}$. Find all of the solutions of the following equation: $$(p-1)!+1=p^{\alpha} \ \ \ ,p>6$$ My attempt We can rewrite the equation as follows: $$(p-1)!=p^{\...
3
votes
2answers
79 views

Given $(2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_rx^r$ Find $\frac{a_7}{a_{13}}$

Given $$(2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_rx^r$$ Find Value of $\frac{a_7}{a_{13}}$ My try: I assumed $A=2x^2$,$B=3x$ and $C=4$ Then we have the following cases to collect coefficient of $x^7$: ...
0
votes
0answers
53 views

Number of positive integers $(m,n)$ [duplicate]

Find the number of positive integers $(m,n)$ for which $m \choose n$$=1984.$ $1984=2^6\times31$ I wrote down the expression for the binomial coefficient but could not deduce anything useful from the ...
1
vote
1answer
30 views

Sum of fraction of factorials should equal zero or one

For this exercise, I need to prove that the following sum $\sum\limits_{i=0}^{n} \dfrac{(-1)^{n-i}(k+i)!}{(n-i)!(i!)^2}$ is equal to zero for $k = 0,1,...,n-1$ and one when $k=n$. I already tried ...
0
votes
1answer
51 views

How to find the number of trailing zeros in a factorial?

E.g you have to find the number of trailing zeros of 28! in base 5. How would you go about it without writing 28! in base 5? Is there a certain step by step method? I already wrote 28! as a product ...
2
votes
1answer
34 views

Sum of product of Stirling numbers

We have for $n>0$, $k>0$ $$\sum\limits_{j=1}^{\min(n,k)}(j!)^2{n\brace j}{k+1\brace j+1}=\sum\limits_{j=0}^{\min(n,k-1)}j!(j+1)!{n+1\brace j+1}{k\brace j+1}$$ How can we prove it?
1
vote
0answers
40 views

Calculating factorial in terms of certain powers

Factorial $n!$ is defined to be$$n!=1·2·3·4\cdots n$$Let's choose base $2$ to represent $n!$, thus$$n!=2^0·2^1·(2^1+2^0)·2^2·(2^2+2^0)·(2^2+2^1)·(2^2+2^1+2^0)\cdots$$Of course, it is just binary ...
2
votes
1answer
38 views

Proving the summation of a double factorial infinite series.

$$ \sum_{n=0}^{\infty }\frac{(-1)^{n}((2n-1)!!)^2}{(2n)! (2^{2n})} = \frac{2}{\sqrt{5}} $$ I came across this summation through some other work, came across the solution as part of a function, but I ...
0
votes
1answer
24 views

Remainder theorem of factorials and powers

I took a competitive exam and there was this question If $7^n$ divides $68!$ Then what is the greatest value of $n$? Can someone tell me how to find the answer.
1
vote
1answer
33 views

Confusion regarding the scope of multiplying a factorial by a constant $30(3k)!$

I've stumbled upon this factorial, which is written exactly in this way: $$30(3k)!$$ As simple as this may seem, I honestly don't know how to interpret this. Does the factorial encompass the entire ...
1
vote
5answers
66 views

Find value of n if ${(n-1)! \over (n-3)!} = 30$

So here's my attempt : $${(n-1)(n-2)(n-3)! \over (n-3)!} = 30$$ $${(n-1)(n-2)} = 30$$ then $$n-1 = 30$$ or $$n-2 = 30$$ then $$n = 31$$ or $$n = 32$$ but my textbook says $$(n-1)(n-2) = 6 * 5$$ ...
9
votes
1answer
362 views

Scary looking limit with an elegant answer

This problem was posted half a year ago by Pierre Mounir on a Facebook group and until now it received no answers. Since most of his problems that I saw were amazing I can bet this one it's worth the ...
3
votes
0answers
21 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
3
votes
2answers
124 views

Does digit $6$ always lead to $\ 25921=161^2\ $?

Consider prime numbers with the property that the product of the factorials of the digits plus $1$ is a perfect square, for example the prime $$30241$$ leads to the square $$3!\cdot 0!\cdot 2!\cdot 4!\...
1
vote
2answers
66 views

How do I solve this combinatorics problem with conditions?

I have $N$ lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has $N-1$ ...
1
vote
2answers
15 views

Subset of combinations in larger set

I am a biologist and not a real mathematician. Hence some of the answers featured here are sometimes too complicated. My question is: I have set of 8 genes named PBX1,ESX1,PIM1,HBB,HBG,BCL11A,KLF4,...
2
votes
1answer
47 views

What is the intermediate step in this negative binomial proof?

On Slide 47 of these slides, there is a formula stating $$ s_k(t) = - \sum_{j=0}^{k} s_j(0) \sum_{n=j}^{k} \frac{e^{-\lambda_n t} \prod_{m=j}^{k-1} \lambda_m }{\prod^{k}_{m=j, m \neq n} (\lambda_m - \...
1
vote
3answers
65 views

Determine $N$ of series $\sum_{n=1}^{N}\frac{n^n}{(2n+1)!}$ so that it differs from the actual sum by less than $\frac{1}{200}$

I can establish that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{n!}$$ But $$\sum_{n=1}^{+\infty}\frac{n^n}{n!}$$ diverges (by the ratio test). And even if it converged I wouldn't have ...
1
vote
3answers
78 views

Distribute 14 books to 2 people so that each has at least 3 books.

I've been trying to solve a problem, but the solution I got looks extrmely unlikely to be right. Perhaps some can point where I'm wrong. The problem is the following. We have $14$ different books. We ...
3
votes
2answers
151 views

In how many ways can 7 women, 10 men sit at table such that no woman sits besides another?

We have $7$ women and $10$ men; they sit at a table. I've been trying to solve how many ways can they sit excluding the case of women sitting next to each other. My reasoning was the following: I ...
0
votes
2answers
43 views

Given positive integers a and b, find values for a and b if a! * b! = a! + b! [closed]

Given positive integers a and b, find values for a and b if a! * b! = a! + b! I have no clue where to start and would appreciate some help, thanks!
0
votes
1answer
28 views

How to show that intermediate values of $\binom{n}{\lfloor n/2 \rfloor }$ are all integers that does not exceed $\binom{n}{\lfloor n/2 \rfloor }$

Let n be a positive integer, and assume that j is a positive integer not exceeding $n/2$. Show that in $n \cdot (n-1)\cdot (n-2)\dotsm(n-j+1)\,/\,j!$, if one alternates the multiplications and ...
2
votes
2answers
88 views

Why $ \lim_{n\rightarrow \infty} \frac{n!}{n^{k}(n-k)! } =1 $?

I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable. Consider the binomial ...
0
votes
1answer
20 views

Factoradic Operations

Recently I have been creating a program that uses Wilson's theorem to solve for prime numbers, but the main disadvantage of that system is that most computers have the 64 long limit, which comes out ...
6
votes
1answer
128 views

The behaviour of $\operatorname{Im}(!n)$

What's going on with the behaviour of the subfactorial's imaginary part? Background: Out of curiosity I tried to construct some recurrence relations using the Pochhammer symbol and out of those came ...
1
vote
0answers
29 views

How limitations affect the number of possible pandigital numbers

First, I understand that the definition of "pandigital" can vary, somewhat, so for the purposes of this question, here's the definition we'll use: "Positive and whole numbers that include each ...