Derangement

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Derangement

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160 posts

#1 h Jan 16, 2025, 2:54 PM

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Derangement of 8 things,,, how to get the number by hand in a short time? Is there a faster way of calculating the formula?

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#2 h Jan 29, 2025, 3:28 PM • 3 Y

Y by Teodormath, youochange, asadova_fatima

:maybe:

Dn = n!/e
But I recommend the well-known formula.

This post has been edited 1 time. Last edited by Ali_369, Jan 29, 2025, 3:28 PM
Reason: it is best

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#3 h Jan 31, 2025, 6:00 AM

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e=2.71828182?

This 'e'?

Thanks!!

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560 posts

#4 h Jan 31, 2025, 7:59 AM

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youochange wrote:

e=2.71828182?

This 'e'?

Thanks!!

Yes. Round to the smaller integer.

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996 posts

#5 h Jan 31, 2025, 8:31 AM

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Wait, how do you obtain this version with $\lfloor\frac {n!}e\rfloor ?$

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#6 h Jan 31, 2025, 9:00 AM

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There is a tiny error in the expression. For odd $n$ s, it should be $\lfloor\frac {n!}e\rfloor$ , but it should be $\lceil\frac {n!}e\rceil$ for all even $n$ .

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70 posts

#7 h Jan 31, 2025, 1:22 PM • 1 Y

Y by youochange

$\boxed{D_n=\left\lfloor \frac{n!}{e}+\frac{1}{2}\right\rfloor}$ For a proof note that: $|{\frac{n!}{e}-D_n}|=n!|\sum_{k\geqslant n+1}{\frac{(-1)^k}{k!}}|\leqslant \frac{1}{n+1}$ and $D_n\equiv (n+1)\mod 2$

This post has been edited 1 time. Last edited by MS_asdfgzxcvb, Jan 31, 2025, 1:25 PM

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#8 h Feb 1, 2025, 2:37 AM

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SBLNuclear17 wrote:

There is a tiny error in the expression. For odd $n$ s, it should be $\lfloor\frac {n!}e\rfloor$ , but it should be $\lceil\frac {n!}e\rceil$ for all even $n$ .

Oh, I didn't know that

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#9 h Feb 3, 2025, 6:20 PM • 2 Y

Y by youochange, asadova_fatima

youochange wrote:

e=2.71828182?

This 'e'?

Thanks!!

Yes it is.No mention

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#10 h Apr 8, 2025, 7:59 AM

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(8! - 8!/1!)+(8!/2! - 8!/3!)+(8!/4! - 8!/5!) + (8!/6!-8!/7!) + 8!/8!
Bruh!
I don't know any faster way to do that!
This is really painful during regional contest only 60 minutes to do 8 question!
In this type of question I make silly addition, multiplication mistake every time!

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