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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3answers
275 views

Why factorials when divided by factorials less than the number have a remainder 0?

Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y2!\cdots y_n!} $as long as Summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the ...
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1answer
84 views

Properties of the $\text{lcm}(1,2 ,… n)$ function

I was thinking the other day about the following function - a sort of prime factorial: $$f(n) = \text{lcm}(1,2,\cdots,n) $$ Does this function have a name? Does it have any interesting properties ...
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4answers
55 views

Is there any number which $n!$ is lower than $2^n$ or same?

I interested in this question. how many numbers meet this condition? I think a few of them meet this but I want a proof for this. also I'm not very pro in mathematics.
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1answer
52 views

how is a factorial fraction equal to the product notation

how is the $\prod$(2k-3) from k=2 to n equal to : ${(2n-3)!\over 2^{n-2}(n-2)!}$ where n $>=$ 2 i know that the (2n-3)! is equal to the product of 2k-3 from k=2 to n but I can't figure out the ...
2
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1answer
53 views

Solving a permutation/Combination equation

please help check if this would the correct way to solve this: $^nP_2 = ^{n+1}C_3$. I want to solve for $n$. theoretically, I was thinking that: $^nP_k = k!\times ^nC_k $ hence: ...
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1answer
59 views

Constant distribution in the factorial 2n!

a very simple question: how does 2 distribute in the factorial $2n!$ ? $2((n)(n-1)(n-2))$ would be treating the expression as $2(n!)$, which is different from: $(2n)(2n-1)(2n-2)$ which is ...
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14answers
4k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
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1answer
33 views

Homework - algebra, find constants

The question is as follows, I think I solved it partially: Show that there are $a,b$ real positive numbers such that $an^7 \leq \frac{n!}{7!(n-7)!} \leq bn^7$ $7\leq n$ my solution for b: ...
2
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2answers
112 views

Prove that $n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}$

Prove that for all $n \in \mathbb{N}$, $$n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}.$$ Thanks in advance.
2
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2answers
128 views

Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$

Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$ is the question I am struggling with. I started by saying: $(2n)!=2n(2n-1)(2n-2)(2n-3)...3*2*1$ But then I'm stuck.
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4answers
71 views

A question about sums and factorials

Consider the sum $S=x!+\sum_{i=0}^{2013}i!$, where $x$ is a one-digit nonnegative integer. How many possible values of $x$ are there so that S is divisible by 4?
3
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1answer
54 views

I don't how do you call this, but please help me. Calculate the result of…

How to calculate the result of: $$\frac{2012(1!)}{3!}+\frac{2012(2!)}{4!}+\frac{2012(3!)}{5!}+...+\frac{2012(2010!)}{2012!}$$ What is the theory used? Sequence of number? Please help me understand ...
4
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1answer
184 views

Can we determine $A= 1!+2!+3!+…$'s digits starting from last?

After reading a bit about p-adic numbers, I came up with an idea. We know that for every natural number $k$, there exists a natural number $n$ so that for every $m>n$, there are at least $k$ zero ...
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1answer
2k views

Proof by induction Involving Factorials

My "factorial" abilities are a slightly rusty and although I know of a few simplifications such as: $(n+1)\,n! = (n+1)!$, I'm stuck I have to prove by induction that: $$\sum_{i=1}^n\frac{i-1}{i!} = ...
2
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1answer
147 views

Solving the 'insert any number of operators' problem

I recently came across a rather simple problem: express the number 2008 using thirteen zeroes (yes, zeroes). The solution is, obviously, to use the fact that $0!=1$ and consequently attempt to use up ...
1
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1answer
69 views

Understanding an Approximation

I am reading the paper A Group-theoretic Approach to Fast Matrix Multiplication and there is an approximation in the paper I don't fully understand. In the proof of Theorem 3.3. it is stated that $$ ...
2
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2answers
169 views

Limit Involving Factorials

How would you go about calculating $$ \lim_{x \to \infty} \frac{x!}{(x - k)!} $$ for some constant $k > 0$?
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1answer
2k views

Mathematical Induction Factorials, sum r(r!) =(n+1)! -1 [duplicate]

How do I prove that $$\sum\limits_{r=1}^{n} r(r!) = (n+1)!-1$$ I was able to get to factor: $LHS = k(k!) + (k+1)(k+1)!$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, RHS = (k+2)! -1$
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1answer
55 views

Computation of $n$-th order difference of falling factorial

I was reading a difference equation textbook and came across a problem. The question asks to compute ${\Delta}^nt^{\underline3}$ for $n=1,2,3,...$, where $t^{\underline3}$ is the falling factorial ...
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1answer
334 views
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1answer
1k views

Using the Squeeze Theorem in Sequences

My textbook has an example that says "Show that the sequence {${c_n}$} $= (-1)^n \frac{1}{n!} $ " converges, and find its limit. It tells me that I must "find two convergent sequences that can be ...
3
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1answer
175 views

Asymptotics for sums involving factorials

This question is rather general, but I have recently encountered the following situation in a variety of different settings. Let us suppose that we are given a complicated sum involving factorials ...
3
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0answers
71 views

Which one is bigger? $2011!$ or $1006^{2011}$ [duplicate]

I want to know Which one is bigger ? $ 2011! \text{ or } 1006^{2011} $. Any proof with logic is most appreciable.
4
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2answers
961 views

Proof that the gamma function is an extension of the factorial function

I've already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And ...
0
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3answers
96 views

Large factorial division

If I want to divide $n!$ by $c^x$ but without simply inputting all in a calculator, what would be the best way to do so? Some example: There are $25!$ atoms on a table. Each second $11^9$ are swept ...
4
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1answer
98 views

Congruences with prime number and factorial

Prove that if $p\equiv 1 \pmod{4}$ is a prime number and $$x\equiv \pm \left(\frac{p-1}{2}\right)! \pmod{p}$$ then $x^2\equiv -1 \pmod{p}$ I think Wilson's theorem will come in handy here, used ...
2
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2answers
61 views

A Better Way to Solve this Factorial Problem?

I had a problem that asked me to find which of the following is larger: ${2013 \choose 500}$ or ${2013 \choose 1500}$ Beneath is my proof. I think it is correct (though your verification and ...
1
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0answers
95 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
5
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1answer
2k views

Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq ...
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1answer
158 views

Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
2
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3answers
67 views

How to solve the inequality $n! \le n^{n-2}$?

The inequality is $n! \le n^{n-2}$. I used Stirling's approximation for factorials and my answer was $n \le (e(2\pi)^{-1/2})^{2/5}$ but this doesn't seem right. Any help would be much appreciated.
1
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1answer
161 views

Double factorial identity

Does anyone know a strategy for proving $$ 2\cdot(2k-3)!!=\sum_{i=1}^{k-1}(2i-3)!!(2(k-i)-3)!!\binom{k}{i} $$ for $k\geq 2$? Note that $(-1)!!=1$. Hints would be most appreciated. Full solutions not ...
3
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4answers
393 views

Is $n \choose k$ defined when $k < 0$? What about $n < k$?

I know that ${n \choose 0} = 1$, and this makes sense to me based on my understanding of combinatorics. But what about ${n \choose -1}$? My instinct is that this is undefined, since it is equivalent ...
3
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2answers
50 views

Find the antiderivative of a function with a finite series and factorials

If $n\in\mathbb{N},s\leq n$, I know that $$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$ I would like to find a similar formula: is there a function $f(t)$ such that $$ \int_0^1 f(t) ...
1
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2answers
95 views

Prove $ \sum_{1\leq k < n} k^{\underline{m}}=\frac{n^{\underline{m+1}}}{m+1} $ by induction on $m$

I want to prove by induction the following sum: $$ \sum_{1\leq k < n} k^{\underline{m}}=\frac{n^{\underline{m+1}}}{m+1} $$ but induction should be on $m$. Any hint will be helpful. EDIT: $ ...
3
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3answers
249 views

Proving that $n!≤((n+1)/2)^n$ by induction

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): ...
4
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3answers
74 views

Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that$\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
6
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1answer
1k views

Is the sum of factorials of first $n$ natural numbers ever a perfect cube?

If $S_n = 1! + 2! + 3! + \dots + n!$, is there any term in $S_n$ which is a perfect cube or out of $S_1$, $S_2$, $S_3$, $\dots S_n$ is there any term which is a perfect cube, where $n$ is any natural ...
0
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1answer
105 views

Summation of reciprocal of Product of Factorials.

How can this summation be evaluated: $${∑ {1\over {a_1!a_2!....a_m!}}}$$ Where $$a_1+a_2+.....+a_m=n$$ Also $a_i !=n $ and $m<n$.
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1answer
69 views

Canceling factorials and exponentials in sum

I'm trying to understand the following the proof. I want to show that $$E\left[\frac{1}{X+1}\right] = \frac{1}{(n+1)p}(1-(1-p)^{n+1})$$ The proof goes like this: $$ \begin{align} ...
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1answer
58 views

Is this an accurate way to represent n! using Π?

I recently learned of the $\Pi$ symbol, and was wondering if the following is an accurate way to represent $n!$: $\Pi_{i=0}^{n-1} n - i$
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2answers
132 views

How to estimate the size of a ratio with very large factorials?

I want to estimate the size of the following ratio: $$\frac{10^{18}!}{10^{14}!\ 10^4!}$$ Since I don't have an idea how to simplify it and no CAS is able to handle numbers of this size, I am at an ...
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2answers
255 views

What is the largest n for n!< 1000?

This is simple factorial equation question. How do you find the largest n satisfying n! < 1000? (Edit) Actually, I want to find some other logic other than brute force. For example, How about ...
3
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2answers
2k views

the nth root of n!?

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
1
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3answers
166 views

Integer ordered pairs $(x,y)$ for which $x^2-y!$…

[1] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2001$ [2] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2013$ My Try:: (1) $x^2-y! = 2001\Rightarrow x^2 = ...
2
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1answer
391 views

How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10} $$ Can ...
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2answers
55 views

How does factorial result in the computation of possible orderings?

For 3 characters to find the ways of putting them in order: ABC, ACB, BAC, BCA, CAB, CBA $= 3! = 3\times2\times1 = 6$ When trying to find ways to put an object in order ...
0
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1answer
94 views

Theory behind multiplication & combinations?

If with the Binomial Coefficient we try to find the possible combinations $\binom{n}{k}$ where $n$ is equal to $k$ what is the theory behind factorial resulting in the correct solution? E.g. ...
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1answer
47 views

Prove that $\frac{n!}{(n-k)!} = n^{\underline{k}}$

I'm having some trouble proving the relation $$\frac{n!}{(n-k)!} = n^{\underline{k}}$$ Do you have to get into using gamma functions in order to prove this rigorously? Also, wikipedia seems to ...
2
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1answer
74 views

Counting permutations, with additional restrictions

There are 10 slots and some marbles: 5 red, 3 blue, 2 green, how many ways can you fit those marbles into those slots? Those marbles fit in 10!/(5! 3! 2!) ways ...