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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proof that $n! \leq (\frac{n+1}{2})^n$ [duplicate]

How do I prove the following statement using induction? $(n \in \mathbb{N})$ $$n! \leq (\frac{n+1}{2})^n$$ So for $n = 1$ it is true, since $1 \leq (\frac{2}{2})^1$. So now: Assumption: $n! \leq (\...
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2answers
38 views

Proving some inequalities with factorials

I'm having some difficulty proving a few inequalities with factorials, and I was hoping someone could point me in the right direction. The first one is for $k\in\mathbb{N}$ and $0\leq t\leq 1$, that $...
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2answers
108 views

Induction proof of exponential and factorial inequality

I'm trying to find a proof for the following statement, using mathematical induction: $$ (\forall n\in \mathbb N-\{0\}) n^n \ge n! $$ But I always get to a dead-end. I've done the basis step, for $...
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1answer
89 views

Selecting a bag of six donuts from eight varieties

I am not really understanding how to solve this set of questions which are involving premutations and combinations, can someone explain question (a) or more? 4.4 How many ways are there to choose a ...
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3answers
41 views

Computing binomial coefficients

Compute from Left-Side: $$ {n \choose p} {n-p \choose k-p} = {n \choose k-p \quad p \quad n-k } = {n \choose k} {k \choose p}$$ So i started computing until: $$ ={n \choose p} {n-p \choose k-p}$$ ...
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1answer
38 views

Factorial minimisation

What methods might be appropriate to find the lowest integer bound for $t$ in the following inequality for a given $p$? $(2t+2)! > 2^{p+1}$ The best I can do is by using the inverse gamma ...
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1answer
63 views

Highest prime factor of factorial.

For a program I wrote, I used the property that the power of the highest prime factor of a factorial is always 1. I couldn't find anything about this, but it felt right. I can't prove it. Is my ...
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2answers
33 views

Are these expressions (Gamma function and binomial) identical for $n\in \mathbb Z$

For $n\in \mathbb Z$ and $n\ge 0$, prove that: $$\frac{2\sqrt{\pi}\,\Gamma(\frac{1}{2}+n)}{\Gamma(n+1)}=\frac{\pi}{2^{2n-1}}\binom{2n}{n}$$ I started to prove. We now that $\Gamma(\frac{1}{2})=\...
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3answers
47 views

Proof by Induction (Inequality)

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
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3answers
26 views

Rearranging for $n$ with a factorial

For my maths course I need to prove that $n!/2^n$ tends to infinity as $n$ tends to infinity. For this I have to rearrange $n!/2^n > ∂$ so that it says $n > ...$. How to do this? Thanks in ...
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1answer
32 views

How to factor constants from a ratio of factorials.

Consider: $$ n_k = [\frac{(k-1)(1-\rho)}{1 + (1-\rho)k}][\frac{(k-2)(1-\rho)}{1 + (1-\rho)(k-1)}]... $$ Where $k \geq 1$ and $0 < \rho < 1$. My interpretation is: $$ \frac{(1-\rho)^{k-1}}{(1-\...
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1answer
69 views

Does $625!$ have $156$ zeros at the end?

Someone wrote that $625!$'s last $156$ digits are zeros because $125+25+5+1=156$. If it's true that $625!$ has $156$ zeros at the end, how does "$125+25+5+1=156$" prove it?
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0answers
46 views

Expression for gcd( ( 0! x n! ) , ( 1! x n-1!) , ( 2! x n-2! ) ,…(k!,n-k!) ) [closed]

Can we reduce computation of $\gcd (0!n!, 1!(n-1)!, 2!(n-2)!,\dots, k!(n-k)!)$? I tried and could not get a general expression for it. Please provide an explanation.
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2answers
102 views

Computation of a limit involving factorial $\lim_{n \to \infty} \sqrt[n+1] {(n+1)!} - \sqrt[n] {(n)!} = \frac{1}{e}$

I want to prove the following limit: $$\lim_{n \to \infty} \sqrt[n+1\;] {(n+1)!} - \sqrt[n] {(n)!} = \frac{1}{e}.$$ I searched the forum & found the link here: If $\frac{p_{n+1}}{np_n} \to p >...
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0answers
37 views

Factor out factorial from expression

I have the expression $(k+1)! - 1 + (k+1)(k+1)!$ How do I factor out $(k+1)!$ to achieve the result: $[(k+1)!(k+2)] - 1$? I for the life of me cannot figure this out. Thanks!
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4answers
175 views

What is the limit of $n!\cdot (2n)! / (3n)!$

$$\lim_{n\to\infty}{n!\cdot (2n)!\over (3n)!}$$ Unsure as to whether to try and divide each term by $(3n)!$ Or where to start really
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2answers
169 views

What is the units digit of $\sum\limits_{n=1}^{1337} (n!)^4$?

What is the units digit of $\sum\limits_{n=1}^{1337}(n!)^4$ ? I find 9 but I am not sure.
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2answers
54 views

Proving $n! > n^2$ by mathematical induction [duplicate]

I'm trying to prove that $n! > n^2$ for $n\geq 4$ by use of mathematical induction, but I get to the inductive step and get lost. But I'm struggling with the inductive step as expected.
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1answer
48 views

Relating to “Find $n$, where its factorial is a product of factorials”

After reading the question and various good answers on the post Find $n$, where its factorial is a product of factorials I wonder if $3! \cdot 5! \cdot 7! \cdots (2n+1)!$ would evaluate to a ...
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3answers
138 views

Is $72!/36! -1$ divisible by 73?

Is $\frac{72!}{36!}-1$ divisible by the number 73? I am not getting a clue in which direction should I go, though I did small amount of work by converting the above expression in the below given form ...
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2answers
60 views

Binomial coefficients inequation problem

Can anyone help me solve this: $$5\binom{13}{x} < \binom{x + 2}{4}$$ After turning it to factorial I don't know what to do nothing seems to cancel out. $x$ is a positive integer. I end up with this ...
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1answer
140 views

How to calculate the number of Combinations (cPr) of a Permutation (nPr)

Im working on a problem where i have 2 variables A= (1,2,3) B= (3,2,1) I have to calculate the number of permutations, and the number of paths by swaping adjancent numbers until A becomes B For ...
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1answer
102 views

Why is the number of bee routes between $5$ flowers not equal to $5!$

On a BBC program about algorithms, they talked about solving the travelling salesman problem. They used the analogy of a bee hive with five flowers to visit before returning to the hive? They said ...
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2answers
74 views

Determine if the following sequence is convergent and find its limit: $x_n=(n^n/n!)^{1/n}$ [duplicate]

Let we have the following sequence $$x_n=\left(\frac{n^n}{n!}\right)^{\frac{1}{n}}$$ Determine if the following sequence is convergent and find its limit
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1answer
65 views

A committee of five is to be chosen from five men and four women.

A committee of five is to be chosen from five men and four women. How many different committees can be formed if the number of men on the committee is to be greater than the number of women? Attempt ...
2
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2answers
297 views

In how many ways can three boys and four girls occupy seven seats in a row if a. A girl and a boy occupy the end seats…

In how many ways can three boys and four girls occupy seven seats in a row if a. A girl and a boy occupy the end seats b. If the four girls must sit together Attempt: For the part a The ...
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0answers
70 views

Factorial of a large number and Stirling approximation

I'm trying to approximate the factorial of a large number with large precision. I know one can use the the Stirling approximation to do that with the formula: $$\sqrt{2\pi x} \left(\frac{x}{e}\right)^...
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2answers
51 views

Prove that using induction that $\binom22+\dots+\binom n2 = \binom{n+1}2$ [duplicate]

so I have this math problem where I have to prove this using induction. $$\begin{pmatrix}2\\2\end{pmatrix}+\begin{pmatrix}3\\2\end{pmatrix}+\begin{pmatrix}4\\2\end{pmatrix}+\cdots+\begin{pmatrix}n\\2\...
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2answers
77 views

Prove that $(a+1)(a+2)(a+3)\cdots(a+n)$ is divisible by $n!$

so I have this math problem, I have to prove that $$(a+1)(a+2)(a+3)\cdots(a+n)\text{ is divisible by }n!$$ I'm not sure how to start this problem... I completely lost. Here's what I know: $(a+1)(a+2)(...
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1answer
65 views

What's a real world example of double exponential function and factorial function? [closed]

As the title asks. I'm looking to create very fast growing numbers. If there's a better solution than these two please let me know as well.
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3answers
74 views

Prove $\frac{(2n)!}{2^nn!}$ is always an integer by induction.

Hey guys so I have this math question. I have to prove that $\frac{(2n)!}{2^nn!}$ is always an integer by induction where $n$ is a positive integer. This is my approach. First I check the initial case ...
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3answers
98 views

Prove $ 2\cdot5\cdot11\cdot19\cdot23\cdot29\cdot31>3^{16} $

I recently came across the following problem: If $C_n=\frac{1}{n+1}\binom{2n}{n}$ is the $n$-th catalan number, then prove that for all $n\ge 17$: $$ C_n>3^n $$ How the induction step works is ...
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4answers
119 views

prove that $\lim_{n\to∞} \frac{n!}{n^n}=0$ [duplicate]

any sugestions?, by Stolz–Cesàro?. The trouble says that first try with $k<n$ and then for a particular $k<\frac{n}{2}$ I try by definition of $\epsilon$ but it doesn't help
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0answers
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How to know if addition of factorials changes the amount of trailing zeroes?

In this specific puzzle, it is $603!+604!+605!$. If it was $603!*604!*605!$ instead, I would simply count all the occurrences of multiples of 5, 25 and 125, which would be $148$ for $603!$ and $604!$,...
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1answer
70 views

I have a mathproblem :)

I am trying to do an epsilonproof. My expression, let's name it a(n), a(n) = n!/2^n I have compared the ratio of a(n+1) / a(n) , this shows me that n! is growing faster than 2^n. I want to prove that ...
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3answers
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Showing that $x^x/(2x)!=0$ as $x$ approaches infinity

How would I show $$\lim_{x\to\infty} \frac{x^x}{(2x)!}=0$$ I know $x^x$ grows faster than $(2x)!$ So then would I do $$\frac{x^x}{2x(2x-1)(2x-2)(2x-3)\cdots(2x-(2x-1))}$$ But how do I proceed.
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1answer
22 views

Factorial manipulation with proving P(n,k)

I need to prove that P(n,k)=$k*P(n-1,k-1)+P(n-1,k)$ So far I have: $$RHS=k*\frac{(n-1)!}{(k-n)!} + \frac{(n-1)!}{(n-1-k)!}$$ $$=k*\frac{(-1+n)!}{(n-k)!}+\frac{(n-1)!(n-k)!}{(n-1-k)!(n-k)!}$$ $$=k*\...
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2answers
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How do I prove the equality of two sums?

I'm trying to prove the following: $$\sum_{r=0}^{n+1} \frac{n! (n+1)}{r!(n-r)!(n-r+1)} = \sum_{r=0}^n \frac{2n!}{r!(n-r)!}$$ I'm pretty sure that these are equal. How could I go about proving it? ...
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7answers
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Find $n$, where its factorial is a product of factorials

I need to solve $3! \cdot 5! \cdot 7! = n!$ for $n$. I have tried simplifying as follows: $$\begin{array}{} 3! \cdot 5 \cdot 4 \cdot 3! \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3! &= n! \\ (3!)^3 \...
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2answers
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Proving $\sum_{n=0}^N n (n!) = (N+1)!-1$ [closed]

It is a piece of cake via induction, but is there any way to derive the formula?
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3answers
127 views

Find all natural numbers $n > 1$ and $m > 1$ such that $1!3!5!\cdots(2n - 1)! = m!$

Find all natural numbers $n > 1$ and $m > 1$ such that $1!3!5!\cdots(2n - 1)! = m!$ I have been thinking about coming up with some inequalities which would narrow the possible range of pairs $(...
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1answer
53 views

How can I prove a limit involving factorial of a log?

Let's say that I wanted to prove the following problem which is similar to a homework problem. That is, the numerator and denominator are different, but similar enough that you posting an answer would ...
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1answer
33 views

Simplification of factorial

Can $\frac{(N_2-k)!}{(N_1-k)!}$ be expressed using $\Delta N = N_2-N_1$ as the only variable? It stems from $\binom {N_2-k}{N_1-k}=\frac{(N_2-k)!}{(N_1-k)!(N_2-N_1)!}$. Thanks!
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4answers
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How to calculate $51! \ mod \ 101$?

I tried almost everything I know Even tried to calculate it from Wilson's theorem and what I got was $$(101-50)! \equiv 51! \equiv (101 + 49!)^{-1} mod \ 101$$ I derived it from $$(p-1)! \equiv -1 \ ...
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4answers
175 views

Help solving an induction problem…$2^n < n!$ [closed]

Prove the following by induction. k and n in N (natural number) $n^k < 2^n $ $2^n < n! $ Need hint and help please. Thank you very much.
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3answers
67 views

Limits and factorials

There is the following limit, I would like to calculate: $$\lim_{n\rightarrow\infty} \frac{2n}{(n!)^{1/n}}$$ After the substituion with Stirling's approximation I have got a relatively complex ...
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1answer
355 views

Finding value of unknown in factorial equation

What's the value of $n$ in the following equation? $$2(2n-4)!= (n-4)!(n+2)!$$ I've tried coming up with an equivalent combination and expanding $(n+2)!$ but that didn't get me anywhere.
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3answers
74 views

What is the significance of $0!=1$?

One can derive $0!=1$ by the formula $\frac{n!}{(n-1)!}=n$ simply put $n=1$ and we get $0!=1$ . But a question remains in my mind: what is its physical significance? For example: $2!$ means: in ...
14
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3answers
127 views

$n!$ as product of consecutive numbers

Let $n$ be a positive integer. In how many ways can one write $n!$ as a product of consecutive integers? For example: $4!=1\times2\times3\times4=2\times3\times4$. Here, $2$ possibilities exist. $...
3
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1answer
75 views

Does the equality $ \dfrac{10!}{6!} = 7! $ hold any special (geometric) meaning?

I've come across the following simple, but unexpected equality numerous times accidentally. $$ \frac{10!}{6!} = 7! $$ which is the same as $$1*2*3*4*5*6*7 = 7*8*9*10$$ Does it hold any specific (...