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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79 views

Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
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1answer
35 views

Expressing the coefficients of $(1-x)^{1/4}$ using factorials

From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$ (1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n. $$ However, can I do the same ...
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2answers
37 views

Show that the sequence is not bounded above

I must show that the sequence is not bounded above: $a_n =\frac{n^n}{n!}$, I tried to use proof by contradiction: suppose there is some $k$ such that $a_n\le k$, then $n^n \le kn!$, $n*n*n...*n \le ...
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1answer
58 views

How to calculate a complicated permutation?

I'm writing a play that features a lot of randomization, which will mean that it is different every time, and I'm trying to calculate the number of performance possibilities. It features a number of ...
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0answers
48 views

How can I simplify $\prod\limits_{k=1}^{n}3k+1$?

I've found that I can simplify $\prod\limits_{k=1}^{n}2k-1$ to $\frac{(2\cdot n)!}{2^n\cdot n!}$, for example: $\prod\limits_{k=1}^{4}2k-1=1\cdot3\cdot5\cdot7=\frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\...
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1answer
72 views

Beta function and gamma function

I would like to ask if someone could help me with following equation. \begin{equation} \Gamma(m)\,\Gamma(n) = \int_{0}^{\infty}x^{m-1}e^{-x}\,dx\,\int_{0}^{\infty}y^{n-1}e^{-y}\,dy \end{equation} \...
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1answer
49 views

For which $n$ is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$?

For how many values of $n$, is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$? Further more, is there a way to ...
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1answer
68 views

Wallis integral and gamma function

I would like to ask if anyone would help me to explain how to reach the following relation. \begin{equation} \int_0^1 \left( 1-x^{\frac{1}{p}} \right)^q dx= \frac{p!\,q!}{(p+q)!} \end{equation} If we ...
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0answers
32 views

Factorial for positive fractions

I know that factorial is also represented by Gamma function and we have exact value of factorial of (1/2), can someone help me how to find numerically(approximate) the factorial of positive fractions ...
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2answers
129 views

How to calculate limit as $n$ tends to infinity of $\frac{(n+1)^{n^2+n+1}}{n! (n+2)^{n^2+1}}$?

This question stems from and old revision of this question, in which an upper bound for $n!$ was asked for. The original bound was incorrect. In fact, I want to show that the given expression ...
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2answers
111 views

Show that $n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ for $n\ge 2$

$n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ For very small values of $n$ (i.e. $2\le n\le 6$) the function on the right nicely approximates $n!$ before significantly overtaking it. I don't have ...
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1answer
80 views

Proof by induction for nth derivative

Show the following hold by induction: $$\frac {d^n}{dx^n}\frac {e^x - 1}{x} = (-1)^n \frac{n!}{x^{n+1}} \left( e^x \left(\sum_{k=0}^{n} (-1)^{k} \frac{x^k}{k!}\right) - 1 \right)$$ Proof. It's not ...
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2answers
42 views

Simplifying factorial with different coefficient [closed]

$$\frac{(pn)!}{(qn)!},\quad p\not = q$$ If possible, how could I simplify the above factorial?
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2answers
44 views

Logarithm of factorial equal to sum of logarithm of primes

Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that \begin{align} \log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...
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1answer
78 views

what is the root of this polynomial?

Let $f_n(x)=\prod\limits_{i=1}^n (x+i)-n!=(x+1)(x+2)\cdots(x+n)-n!$ $n$ is a positive integer. What are the roots of the polynomial for a given $n$ except $0$? Or determine the real part of the ...
5
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3answers
109 views

Why do the factorials appear in differences of consecutive powers?

Why do the factorials appear when repeatedly taking the differences of consecutive powers? Or rather why is the $n_{th}$ factorial equal to the $n_{th}$ difference of $(k+1)^{n}-k^n$? I'm having ...
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0answers
35 views

What's a compact way to express the product of terms in arithmetic progression?

There is a notation for the product of the first $n$ positive integers. My question is whether it is possible to express the product of any other non-trivial arithmetic progression with $n$ terms ...
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4answers
299 views

The asymptotic behavior of $n\ln n -n$ [closed]

How do I show that $$\displaystyle\lim_{n\to\infty}\dfrac{n\ln n - n}{\ln n!}=1?$$
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1answer
29 views

Need proof/explanation for a problem involving factorials

Suppose you have a list of integers from 1 to N And say you remove any two numbers, X and Y, from the list of numbers, and add (X + Y + X.Y) to the list If you keep doing this until you have only ...
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2answers
44 views

Product representation of $(2n)!$?

I know that $n!:=\prod_{k=1}^nk$. Would it just be $(2n)!:=\prod_{k=1}^n2k$ or $(2n)!:=\prod_{k=1}^{2n}k$ Any ideas?
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1answer
49 views

Euler Representation of the factorial

I know that Euler proved that $\displaystyle z! = \int_{0}^{1} \left(- \ln\left(t\right)\right)^z dt$ How can this be proven?
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2answers
38 views

Is there a sum of “n+j” terms equivalent to n!

Is there any function so that... $$\sum_{k=0}^{n} f(k) = n!$$ or, $$\sum_{k=0}^{n+j} f(k) = n!$$ where j is any arbitrary integer?
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5answers
37 views

Factorial quotient with same number of factors

I have this limit: $$\lim_{x\to \infty} \frac{n!(3n)!}{((2n)!)^2}$$ How can I resolve it? I know that it results $+\infty$
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2answers
65 views

How to obtain the identity $\sum_k\frac{(m-k)!}{(n-k)!} = \frac{(m+1-k)!}{(m+1-n)(n-k)!}$

Hereby $$\sum_k\frac{(m-k)!}{(n-k)!} = \frac{(m+1-k)!}{(m+1-n)(n-k)!} =: SOL(k)$$ is supposed to be similar to the way of writing antiderivatives. In complete form it should be like this: $$\sum_{k=a}^...
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3answers
47 views

How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
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1answer
39 views

Factorial solve without substitution [closed]

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 ...
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1answer
49 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.
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3answers
107 views

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

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 ...
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3answers
69 views

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

What is the limit of $$\lim\limits_{x \rightarrow 1} \frac{ (x +x^2+x^3+ \cdots +x^n)-n}{x-1}$$
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1answer
34 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 ...
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1answer
102 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
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2answers
48 views

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

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 ...
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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 ...
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1answer
30 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.
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1answer
33 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)...
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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
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1answer
51 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 ...
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0answers
31 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 ...
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0answers
20 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.
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1answer
47 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
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2answers
66 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 = ...
1
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1answer
33 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((...
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0answers
14 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/...
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3answers
48 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}...
4
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2answers
112 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
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1answer
21 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
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1answer
34 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
1
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1answer
31 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
80 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
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0answers
55 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 ...