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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2
votes
1answer
114 views

Factors of ${{x+n} \choose {n}}$

Let $v_p(n)$ be the highest power of $p$ that divides $n$. It seems to me that for any prime $p$, $v_p\left({{x+n} \choose {n}}\right) \le \max\left(v_p(x+1), v_p(x+2), \dots, v_p(x+n)\right)$ Am I ...
2
votes
4answers
434 views

Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$

I'm looking for a way to find this limit: $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$ I think I have found that it diverges, by plugging numbers into the formula and "sandwich" the result. However I ...
2
votes
2answers
276 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}.$$ I am unable to do this one. ...
2
votes
4answers
2k views

Factorial expressed in terms of two other factorials

Can the factorial of $N$ always be expressed by the sum(addition and subtraction) or the product of two other factorials? Do there always exist integer $A$ and $B$ such that $N! = A! + B!$, or $N! = ...
1
vote
4answers
877 views

Find the remainder when $45!$ is divided by $47$?

Find the remainder when $45!$ is divided by $47$? My approach I am using Wilson's theorem to solve the problem. I reduced the expression into ($47$-$1$-$1$)!/$47!$=$(47$-$1$)/$47$!-$1$/$47!$=-$1$-$...
0
votes
2answers
173 views

Showing $(n+1)^n<e^nn!$ by induction

Show $(n+1)^n<e^nn!$ I know why that would be the case using general knowledge and a bit of substitution but am clueless on how to prove it.
8
votes
3answers
203 views

Inductive Proof that $k!<k^k$, for $k\geq 2$.

Call $P(k): k!<k^k$, for $k\geq 2$ Test it out with 2, and it's true ($2<4$). Assume that $P(k)$ is true for some $k\geq2$. Then show that $P(k+1)$ is true. $P(k+1): (k+1)!<(k+1)^{k+1}$ ...
6
votes
3answers
18k views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
5
votes
3answers
636 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
5
votes
1answer
4k views

$n$ choose $k$ where $n$ is less than $k$

I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, ...
5
votes
2answers
95 views

Is there a close form for $g(a,b,n)=\sum\limits_{k=0}^{n}\binom{n}{k}\frac{1}{ak+b}$?

We can be sure, that for $a>0$, $b>0$ $$f(a,b,n)=\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{ak+b}=\frac{(an)!^{(a)}}{(an+b)!^{(a)}}$$ where $(an+b)!^{(a)}$ denotes multifactorial: $(n)!^{(1)}...
5
votes
3answers
347 views

Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)

I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate: $$\lim_{x\to-3}\frac{...
5
votes
3answers
171 views

How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$

$$\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$$ Can someone show why this estimate holds true? I tried quite a bit but couldn't really find a way to approach this. WolframAlpha says it is true ...
5
votes
3answers
4k views

What is the definition of $2.5!$? (2.5 factorial)

I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater ...
4
votes
3answers
947 views

What algorithms exist to quickly compute the inverse factorial?

I'm interested in algorithms to quickly compute the inverse factorial. I've noted that large factorials have a unique number of digits. How can I use this fact to quickly compute the factorial? Is ...
4
votes
4answers
16k views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
4
votes
4answers
146 views

Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

Is there a proof for the following identity that only uses the definition of the (generalized) binomial coefficient and basic transformations? Let $n$ be a non-negative integer. $$\binom{2n}{n} = (-4)...
3
votes
1answer
187 views

Is there an equivalent to the Bertrand's postulate between factorials and primorials?

As the title explains, I am trying to know if there is a definition about the upper limit to find the first primorial $p_i\#$ (following the definition at OEIS) existing after a given factorial $n!$ (...
3
votes
3answers
174 views

Expanding $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial

I want to expand $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial, where $\Gamma$ is the gamma function. For $k\in\mathbb{N}$, it can be "simplified" as $$\frac{\Gamma(n)}{\Gamma(n-k)}=(n-1)(n-2)(n-3)...
3
votes
4answers
760 views

Evaluate $\lim_{x\to \infty}\ (x!)^{1/x}$

Here's the problem... $$\lim\limits_{x\to \infty}\ (x!)^{1/x}$$ I've deduced the answer to this is $\infty$, but haven't exactly shown that. I'm getting $\infty^0$, so did both tricks where you ...
3
votes
4answers
94 views

How to apply induction to this formula?

I want to justificate following equation: $$\sum_{k=0}^n \frac{(-1)^k}{k!(n-k)!}\frac{1}{2k+1} = \frac{2^n}{(2n+1)!!}$$ I calculated the both sides for $n$ from 1 to 10 and it was true. How the ...
2
votes
4answers
2k views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
2
votes
4answers
195 views

Evaluate $\displaystyle \lim_{n\to \infty}\sqrt[n]{n!}$

I am trying to evaluate the $\lim(\sqrt[n]{n!})$ using 2 theorems (2 proofs) Theorem 1: Let $\{c_n\}$ be any sequence in $\mathbb{R}^+$. Then, $\displaystyle \underline{\lim}\frac{c_{n+1}}{c_n}\leq \...
1
vote
2answers
3k views

Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$ [closed]

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$
1
vote
2answers
826 views

Estimate of n factorial: $n^{\frac{n}{2}} \le n! \le \left(\frac{n+1}{2}\right)^{n}$

on our lesson at our university, our professsor told that factorial has these estimates $n^{\frac{n}{2}} \le n! \le \left(\dfrac{n+1}{2}\right)^{n}$ and during proof he did this $(n!)^{2}=\...
1
vote
0answers
57 views

Verifying an intuition about a sequence of consecutive integers

Let $x>0$ be the first integer in a sequence and $n>0$ be the number of consecutive integers in the sequence. For example, if $x=12,n=3$ the sequence would be $\{ 12,13,14\}$ Let $v_p(x,n)$ be ...
-4
votes
1answer
659 views

Find the number of trailing zeros in $n!$. [closed]

Can anyone give me a generalized way to find the number of zeroes trailing at the end of $n!$ ?
8
votes
4answers
2k views

How many consecutive composite integers follow k!+1?

I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's ...
6
votes
2answers
513 views

Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
5
votes
9answers
658 views

How can one prove $\lim \frac{1}{(n!)^{\frac 1 n}} = 0$?

I have tried bounding the terms by $\dfrac 1 {2^{\frac 1 n}}$, but this clearly cannot be made as small as possible.
5
votes
8answers
1k views

How to prove that $\lim\limits_{n\to\infty} \frac{n!}{n^2}$ diverges to infinity?

$\lim\limits_{n\to\infty} \dfrac{n!}{n^2} \rightarrow \lim\limits_{n\to\infty}\dfrac{\left(n-1\right)!}{n}$ I can understand that this will go to infinity because the numerator grows faster. I am ...
4
votes
2answers
320 views

Proving $r!$ divides the product of r succesive positive integers [duplicate]

I have to prove the following theorem: Prove that the product of $r$ consecutive positive integers in divisible by $r!$ I am having a hard time getting a generalization down for the full set of ...
4
votes
2answers
4k views

Factorial Moment of the Geometric Distribution

I am trying to caclulate the Factorial Moment of the Geometric Distribution #2 with parameter $p$. Therefore I set $\Omega = \mathbb{N}_0$ and have by using the pochhammer symbol and setting $q=1-q$ ...
4
votes
3answers
234 views

Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$

Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I ...
4
votes
2answers
2k views

How many bits are in factorial?

I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values. In other words I need ...
3
votes
3answers
387 views

Proof by induction: inequality $n! > n^3$ for $n > 5$

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 ...
3
votes
2answers
161 views

Limit $\lim_{n \to \infty} \frac{n!}{n^n}$ [duplicate]

$$\lim_{n \to \infty} \frac{n!}{n^{n}}$$ Attempt: $$n!=n\left( n-1 \right)\left( n-2 \right)...\left( n-\left( n-1 \right) \right)=n^{n}+...$$ $$\lim_{n \to \infty} \frac{n!}{n^{n}}=\lim_{n \to \...
3
votes
4answers
222 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$ [duplicate]

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!}$ So I proved the base case where $n=1$ and got $\frac{1}{2}$...
2
votes
3answers
94 views

Is it correct to evaluate combinations of two as sum?

I've recently had a look to a problem solution that puzzles me. ANd Ican't make up my mind in order to explain myself why it seems actually to work. Consider having N unique numbers and find all ...
2
votes
3answers
1k views

In how many ways can you select one of the two but not both?

For this question: A committee of three boys and three girls is to be selected from a class of 14 boys and 17 girls. In how many ways can the committe be selected if: a.) Ana has to be on the ...
2
votes
0answers
63 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
2
votes
2answers
1k views

Does series with factorials converge/diverge: $\sum\limits_{n=1}^\infty \frac{4^n n!n!}{(2n)!}$?

$$\sum_{n=1}^\infty {{4^n n!n!}\over{(2n)!}}$$ I tried the ratio test but got that the limit is equal to 1, this tells me nothing of whether the series diverges or converges. if I didn't make any ...
2
votes
3answers
2k views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$, prove the equality $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ Here, $x^{\underline{j}}$ denotes a falling ...
2
votes
1answer
145 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
1
vote
2answers
105 views

Factorial sum estimate $\sum_{n=m+1}^\infty \frac{1}{n!} \le \frac{1}{m\cdot m!}$

Prove that: $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$ I have tried induction on $m$ but it does not work very well. Any suggestion?
1
vote
2answers
109 views

Prove $1(1!)+\dots+n(n!) = (n+1)!-1$ using induction

So I'm trying to prove this statement (through induction): $$1(1!)+2(2!)+\dots +n(n!)=(n+1)!-1$$ But I'm confused with the inductive step here: $$(n+1)!-1+[(n+1)(n+1)!] = (n+2)!-1$$ What do I do ...
1
vote
3answers
136 views

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$ So far I have: Base case (n = 1) = $8^{1} | (4(1))!$ = $8 | 24$ which is true. Induction Step: $8^{n + 1} | (4(n + 1))...
1
vote
3answers
465 views

Proof of $\gcd(n!+1, (n+1)!+1)=1$?

I guess the canonical proof to this problem is attained using the Euclidean Algorithm (I've seen some posts like this already). I came with this proof, based on gcd definition and some divisibility ...
1
vote
1answer
107 views

Proof by strong induction combinatorics problem: $1(1!) + 2(2!) + 3(3!) + \dots + n(n!) = (n+1)! - 1$

$1(1!) + 2(2!) + 3(3!) + \dots + n(n!) = (n+1)! - 1$ How do we prove this by strong induction? I know how to do it with weak induction, but how would strong induction work with this problem?
1
vote
1answer
75 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...