# 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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### Prove the given inequalities. [duplicate]

Prove the following inequalities,$$(n!)^2 ≤ n^n(n)! <(2n)!$$ My attempt I proved one of the inequality using mathematical induction. To prove - $(n!)^2 ≤ n^n(n)!$ For $n = 1$, LHS $≤$ ...
2answers
103 views

### $34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$

There was a number theory question that I have to do for homework. $34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$ I know $b=0$ because $10^7\big|34!$ only. But how can ...
1answer
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### Prove that $n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$ [closed]

Prove that if $s_n=1^1+2^2+3^3+\cdots+n^n$ then $$n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$$ (it holds for $n$ larger than $2$). I want to prove ...
1answer
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### Four elevators stop on floors $1$ to $8$; a separate elevator stops on $1$ and $4$. Suppose you want to get from $1$ to $4$ …

I recently encountered an interesting math problem at the hospital. The hospital had two elevator banks. The first one had four elevators that went to any of the floors, 1 through 8. While the second ...
2answers
98 views

### How does $a^{a^n}$ compare with $n!$ asymptotically?

I am learning Asymptotic complexity of functions from CLRS. I know that exponentiation functions like $a^n$,$(a>0)$ are faster than $n!$ But what about $a^{a^n}$ vs $n!$ How do they compare? A ...
1answer
49 views

### 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 ...
1answer
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### How do we obtain the following: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$

I saw the following equality in an informal proof: $\lim_{n \to \infty}\frac{\ln{n^n}}{\ln{n!!}}=\lim_{n \to \infty}\frac{\ln{n^n}-\ln(n-2)^{n-2}}{\ln{n!!}-\ln(n-2)!!}$ I did not understand it and ...
3answers
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1answer
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### Prove that $~~~~~a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+…+a_m\right)! ~~~~~~~~~\forall ~~~a_1,a_2,…,a_m\in N$

Show that $$a_1!a_2!\cdots a_m! \mid \left(a_1+a_2+...+a_m\right)! \forall a_1,a_2,...,a_m\in N$$ Case 1 $m=2$. We need to prove $$a_1!a_2!\mid (a_1+a_2)!$$ $a_1+a_2=1$ and $a_1+a_2=2$. It is ...
1answer
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### Approximation of $x!$

I have discovered a new formula that approximates the factorial function. It is more accurate than Stirling’s approximation. How do I go about publishing it and receive the credit for it?
1answer
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### Proving $\lim((n!)(\frac{e}{n})^n) = \infty$ using elementary method

Is there some easy way to show that $\lim_{n\to\infty} ((n!)(\frac{e}{n})^n) = \infty$ as $n \to \infty$? It looks like Wolfram alpha is using some kind of expansion method to numerically compute this?...
2answers
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### how do I do I solve this equation [closed]

(n+1)! = 110(n-1)! I have searched online for a week now and my textbook does not explain how it got the answer: n=10.
1answer
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### solve for the lowest value of $k$ in ${n}\choose {k}$ $\geq$ $x$ , given $x$ and $n$

Let's suppose that $n = 10$ and $x = 500$. I want to find the smallest value of $k$ for which ${n}\choose {k}$ $\geq$ $x$ I can check for all values of $k$ from $1$ to $n/2$ and pick the least one ...
2answers
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### Query regarding a theorem in Number Theory

Please refer to Theorem $8$ in the attached picture. I do not understand the significance of the condition $(n+1)!+k>k$. Is it not always true since the inequation implies $(n+1)!>0$ which is ...
1answer
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3answers
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