# 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.

2,561 questions
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### Maximising $\frac{x^n}{n!}$

Let $x$ be a number such that $x\gt 0$ and $x\in\mathbb{R}$. Is it true that the maximum value of the expression $$\frac{x^n}{n!}$$ occurs for $n\in\mathbb{N}$ where $n=\lceil x \rceil - 1$? If true, ...
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### How to find summation of factorials

I got stuck at the following summation while solving another problem. $$\sum_{k=n}^N \frac{(k)!}{(k-n)!}$$ I expanded the summation but have no clue how to simplify it.
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### Calculator giving weird answer when dividing factorial

I am using a TI-34 MultiView I was trying to divide the following 20!/(17!3!) The answer should be 1140 right? the numerator is 2.43*10^18 the denominator ...
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### Similarity between $e^x$ power series and Gamma function integral?

The power series for $e^x$ is as follows. $$e^{x} =\sum ^{\infty }_{n=0}\frac{x^{n}}{n!}$$ If we define $n! = \Gamma(n+1)$, then we have $$n!=\int ^{\infty }_{0} x^{n} e^{-x} dx.$$ An extremely ...
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### Permutation: How to arrange 12 people around a table for 7?

I want to understand how to arrange $12$ people around a circular table with $7$ chairs. We don't care about the overflow, those people can go to another table. I thought the way to solve the problem ...
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### Show that $x! y! = z!$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$.) [closed]

Show that $$x! ·y! = z!$$ has infinitely many solutions. (Hint: For example, $5! 119! = 120!$) I am stuck on this problem. Within this section we are learning Congruence. So I know it involves ...
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### How to solve $(x!)!+x!+x=x^{x!}$

How to solve this equation $$(x!)!+x!+x=x^{x!}$$ The answer is $3$ . But I have no idea of how to solve it. Thanks for your time.
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### Analytically continuing the product of the first $n!$ to negative numbers?

Analytically continuing the product of the first $n!$ I recently had the following idea to use the below identity: $$(1!2! 3! \dots n!) (12^2 3^3 4^4 \dots n^n) = n!^{n+1}$$ If we focus on the ...
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### Lottery odds question

In this lottery 5 balls are chosen from 1-50. A friend offers me 5 to 1 if I can get one number correct from the 5 chosen, the order it comes up doesn't matter. Is this a good bet to take? To get ...
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### Inequality with logarithmic function

Prove the following inequality for any $n ∈ [2;\infty)$: $$\log_{n!}(\frac{n+1}{2})> \frac{1}{n}$$
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### What are the steps to solve this problem? [closed]

If $\frac{1}{2n!}$ , $(n-2)!$, $(2-n)!$ are the side lengths of a triangle in cm., then what is the numerical value of the area of the triangle ?
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### A subword in a word

Probability So I have been trying to solve this question in probability, but I don't seem to get the correct answer. I am not bad at probability and this seems to be easy one, but I'm just struggling ...
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