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.

15
votes
8answers
19k views

How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$ [duplicate]

It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc... I thought about writing: $$ a(n) = \frac{...
3
votes
0answers
216 views

How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where $\...
0
votes
1answer
94 views

Proving that a binomial coefficient is the sum of two others [duplicate]

I am asked to prove this given $1 \le m \le n - 1$ (homework question): $$\frac{n!}{(m-1)!(n-m+1)!} + \frac{n!}{m!(n-m)!} = \frac{(n+1)!}{m!(n+1-m)!}$$ Which proof technique should I use to solve ...
5
votes
3answers
264 views

Proving $n! > n$ for $n > 2$ using mathematical induction

I have to prove $n<n!$ for all $n>2$ by mathematical induction. I did it as follows. I proved the base case. Then let it be true for $K>2$: $$ K<K! $$ I have to prove, $$ K+1<(K+...
0
votes
1answer
150 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...
6
votes
3answers
2k views

Asymptotics of terms and errors in Stirling's Approximation

I have two related questions. Both are related to the asymptotics of Stirling's approximation, which is why I have included them in the same question. I will separate the questions if it is deemed ...
1
vote
4answers
176 views

False statement proven by induction?: $ n\geq a\Rightarrow n!\geq a^n, n\in \mathbb{N}-\left \{ 0 \right \}$

Can you spot my mistake? I will show the false statement, that $n\geq a\Rightarrow n!\geq a^n, n\in \mathbb{N}-\left \{ 0 \right \}$, with induction For $n=1$ , $1\geq a\Rightarrow 1!\geq a^1\...
2
votes
2answers
415 views

Proof for convergence of a given progression $a_n := n^n / n!$

"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable. (a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$ [...]" I am having problems ...
61
votes
16answers
49k views

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is being ...
19
votes
2answers
15k views

Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
13
votes
5answers
980 views

$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction

I have to show $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction. This is where I am stuck: $$\left( \frac{n+2}{2} \right)^{n+1} \geq \dots \geq =2 \left( \frac{n+1}{2} \right)^{n+1} = \...
2
votes
2answers
834 views

How to prove $n < n!$ if $n > 2$ by induction?

I am stuck with the question below, Prove by mathematical induction that $n<n!$ for $n>2$.
0
votes
2answers
3k views

Simplifying this factorial expression

I know that $\frac{(m-1)!}{(m-n)!(n-1)!} + \frac{(m-1)!}{(m-n-1)!(n)!} = \frac{m!}{(n)!(m-n)!}$, but I am not sure on the intermediate steps. The only solution I am seeing involves finding a common ...
0
votes
1answer
279 views

Simplify a equation containing factorial, summation, and fraction

I really need some help on simplifying this math equation. Please help me reduce it as simple as possible! Thanks in advance! $$\large P_0=\frac{1}{\left[\sum_{i=0}^{M-1} \frac{1}{i!}\left(\frac{\...
1
vote
1answer
615 views

Why is the zero factorial one i.e ($0!=1$)? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles Why does 0! = 1? I was wondering why, $0! = 1$ Can anyone please help me understand it. Thanks.
7
votes
3answers
3k views

Inductive proof for the Binomial Theorem for rising factorials

I want to proove the following equality containing rising factorials $$(x+y)^\overline{n}\overset{(*)}{=}\sum_{k=0}^n\binom{n}{k}x^\overline{k}y^\overline{n-k}.$$ For $n=1$ this equality is ...
23
votes
3answers
14k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
28
votes
2answers
12k views

Is there a way to solve for an unknown in a factorial?

I don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1. $n! = \text {a really big number}$ Ex. $n! = 9999999$ Is there a way to ...
11
votes
2answers
4k views

What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?

$$\lim_{n \to\infty} \frac{n!}{n^n}$$ I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence?
34
votes
1answer
949 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty \left(\frac{n^n}{n!e^n}-\frac{1}{\sqrt{2\...
13
votes
5answers
10k views

Solve by induction: $n!>(n/e)^n$

To Prove : $n! > (n/e)^n$ The question seems easy but it ain't; anyone up for it ?
0
votes
1answer
6k views

Probability, factorials, the mystery of it all

Continuing on with the GRE practice questions I have, I'm confused about how to solve the following types of probability questions. Martha invited 4 friends to go with her to the movies. There are ...
1
vote
1answer
122 views

Combination Problem with a Variable

I have the following problem: $_xC_6$ = $_xC_4$ I expand both sides to: $$\frac{x!}{[(x-6)!]6!} = \frac{x!}{[(x-4)]!4!}$$ Next I multiply both sides by the denominator of the right-hand ...
1
vote
1answer
895 views

Permutation Problem with a Variable

13P5 = 1287(xPx) I simplify the above to: 13!/8! = 1287(x!) The expression on the left simplifies to 154,440. I divide both sides by 1287 to get: 120=x! At this point I'm stumped. Thanks in ...
6
votes
2answers
1k views

How accurately can huge factorials be calculated?

Given a number like $10^{20}!$, what can I, in a reasonable amount of time, figure out about it? Can I figure out how many digits it has, and/or what the first digit is? I found Striling's ...
37
votes
8answers
3k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
4
votes
3answers
88 views

Reducing $\prod \limits_{0 \le j \ne i \le n} \frac{n+1-j}{i-j}$ to $\frac{(n+1)!}{(n+1-i)\cdot i! \cdot (n-i)!}(-1)^{(n-i)}$

How could we show that: $$\prod_{0 \le j \ne i \le n} \frac{n+1-j}{i-j} = \frac{(n+1)!}{(n+1-i)\cdot i! \cdot (n-i)!}(-1)^{(n-i)} .$$ The module suggest we could reduce it by simply writing $$\prod_{...
4
votes
2answers
3k 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 $q=1-q$ ...
3
votes
1answer
907 views

What is the theoretical upper bound of factorion numbers?

Recently I read about factorion numbers. I understood that there are only 4 factorion numbers, but what is the theoretical range in which they can be? Is it $[0, +\infty]$ or a smaller upper range? ...
101
votes
1answer
5k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $...
89
votes
7answers
8k views

Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
11
votes
2answers
437 views

Given $n! = c$, how to find $n$?

I'm dealing with a time-complexity problem in which I know the running time of an algorithm: $$t = 1000 \mathrm{ms} .$$ I also know that the algorithm is upper bounded by $O(n!)$. I want to know ...
3
votes
5answers
510 views

Factorial of 0 - a convenience? [duplicate]

If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the ...
11
votes
5answers
3k views

Prove that $\gcd(n!+1,(n+1)!+1)=1$

I'd like to solve this one similarly to my previous question: Is this a Valid proof for $(2n+1,3n+1)=1$? I did find a somewhat related post that uses a different method: How to show that $\gcd(n! + 1,...
11
votes
5answers
16k views

Why does 0! = 1? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles Why does $0! = 1$? All I know of factorial is that $x!$ is equal to the product of all the numbers that come before it. The product of 0 ...
27
votes
1answer
1k views

Repeated Factorials and Repeated Square Rooting

I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
2
votes
1answer
578 views

Use of algebra and factorials for a question related to proof by induction

$$ \begin{align*} &= (n+1)! − 1 + ( (n+1) · (n+1)! )\\ &= (n+1)! (1+n+1) − 1\\ &= (n+1)! (n+2) − 1\\ &= (n+2)! − 1\\ \end{align*} $$ I'm confused at how the first ...
2
votes
2answers
1k views

Combination vs Permutation?

This idea resulted while I heard an advertisement for Sonic, where they claim to have something like 300,000 different drinks they serve. Essentially, what they are allowing you do to is mix any soda ...
21
votes
5answers
4k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq en\left(\...
37
votes
4answers
2k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
2
votes
3answers
934 views

Arrangement of six triangles in a hexagon

You have six triangles. Two are red, two are blue, and two are green. How many truly different hexagons can you make by combining these triangles? I have two possible approachtes to solving this ...
63
votes
18answers
13k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
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 ...
0
votes
1answer
2k views

Logarithm base 2 and factorials

I'm learning about $\log_2$ for an algorithms class and theres a problem in the book that is confusing me. It asks: Find a formula for $\log_2(n!)$ using Stirling's approximation for $n!$, for large $...
7
votes
1answer
477 views

Is this a new formula?

It's late at night and I'm tired, but I just stumbled across this while doing my homework. Any chance this is new? Or, maybe, did I just somehow transform it and it is still basically the same formula?...
3
votes
1answer
724 views

How do I describe the growth of something that scales by a factorial?

I just wrote a blog post and wasn't sure how to word a particular sentence. Say I have the following function: \begin{equation} f(x) = x^2 \end{equation} Then I can say that the value of f(x) grows ...
8
votes
3answers
10k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$ [duplicate]

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
9
votes
2answers
1k views

Number of zeros not possible in $n!$ [duplicate]

Possible Duplicate: How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? The number of zeros which are not possible at the end of the $n!$ is: $...
44
votes
3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
0
votes
1answer
212 views

Show that $\Pi_{i < j} (v_i - v_j) \le k^{n^2}$ for $1 \le v_1 < v_2 < … < v_n = k$

Everything is in $\mathbb{Z}$. Let $v_1 < v_2 < ... < v_n = k$, and $v_1 = 1$ for $k >> n$. Let $ P = \Pi_{i < j} (v_j - v_i)$. How can I show that $P \le k^{n^2}$? There are $n + (...