# 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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### Permutations excluding repeated characters

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the ...
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### What is longer? Length of $2^n$ or number of $0$ in $n!$?

What is longer? Length of $2^n$ or number of $0$ at the end of $n!$ ? Assume that $n$ is really big, big number... Solution Look at terms of $2^k$: $$2,4,8,16,32,64,128,256,512,1024,2048,4096...$$ ...
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### $n$-th derivative of $x^\alpha$ where $\alpha = m + 1/2$

It is well-known that, for any real $\alpha$ and nonnegative integer $n$ $$\frac{d^n x^\alpha}{dx^n} = \alpha(\alpha-1)\cdots(\alpha - n + 1) x^{\alpha - n}$$ I just found out that the coefficient ...
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### Show that $(a!)^b b! \mid (ab)!$ [duplicate]

Show that if $a$ and $b$ are positive integers then $(a!)^b b! \mid (ab)!$ This is what I have done: The above statement is true for $a=1$ and any arbitrary value of $b$, and also for $b=1$ and ...
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### For What Non-Negative Integer Values of $n$ is $n!\geq 3^n$

How could I solve for the $n$ in this instance using discrete methods or is this something that I have to do by hand/computer? I've seen this problem in inductive proofs but the base case is usually ...
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### Simplifying an alternating sum of a product of factorials

For integers $a$ and $b$, I am curious how to simplify an expression of the form $$\sum_{k=1}^n (-1)^k (a+k)! (b+k)!$$ I assume there is some simplification using properties of gamma and beta ...
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### Distribution of digits across all factorials

Show that the distribution of zeroes across all digits of all n! for $n\in \mathbb{N}$ converges to $\frac{1}{6}$ and hence, $\frac{5}{54}$ for all other digits 1 through 9. In other words, let's ...
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### What about irrationality of $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$? [closed]

I would like to learn more about the behavior of the factorial function or Gamma function, so I decided to compute $$\int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}\mathrm dx.$$ According to Wolfram alpha, ...
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### How do I solve this combinatorics problem with conditions?

I have $N$ lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has $N-1$ ...
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### Computational results for the sequence $n!+{p_n}!+1$ are, well, very very unusual

Peter and I were discussing in a chat room and I thought that it would be nice to test the sequence $$n!+{p_n}!+1$$ for primality. Then I wrote Peter that I expect much of primes in this sequence, ...
Actually I found out that if $n = 2^k$, $k\in Z_{+}$ we can say that it's true. Here's the proof: Let $n = 2^k$, $k\in Z_{+}$ Then $\nu _{2}(n!) = \nu _{2}((2^k)!) = \lfloor \frac{2^k}{2} \rfloor + \... 3answers 61 views ### Cannot find a calculator that can handle 220! x 40000 For a game I am playing, I am trying to calculate the cost of purchasing all the expansion area squares on a level. the first square costs 40,000, each next square costs an additional 40,000 more than ... 3answers 908 views ### What algorithms exist to quickly compute the inverse factorial? I'm interested in algorithms to quickly compute the inverse factorial. I've noted that large factorials have a unique number of digits. How can I use this fact to quickly compute the factorial? Is ... 1answer 49 views ### What is the smallest integer$k$, for fixed$n$(or vise versa), does$2^k! \geq 2^{n-k}$? What is the smallest integer$k$, for fixed$n$(or vise versa), does$2^k! \geq 2^{n-k}$? I tried to find$k$for small$n$so I could get some OEIS hit, but both sides just grow too fast for me to ... 1answer 73 views ### Functions whose Limit is the Factorial Function I want to know examples of functions$f(n)$whose limit is$n!\$ Now, when I say "limit", I don't mean $$\lim_{n \to \infty}\frac{f(n)}{n!}=1$$ (I already know functions like that). I'm referring to ...
Could someone show me the steps to get from $$(n+1)(n+1)!+(n+1)!-1$$ to $$(n+1)!\big((n+1)+1\big)-1$$ I'm having trouble understanding the simplification. Thank you!!!