Martians and Human with Circular table

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Martians and Human with Circular table

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Source: Bangladesh National Mathematical Olympiad 2016

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Olympus_mountaineer

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Olympus_mountaineer

#1 h Feb 26, 2019, 6:22 PM • 2 Y

Y by Adventure10, Mango247

Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling.

(a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line?

(b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?

This post has been edited 1 time. Last edited by Amir Hossein, Jan 15, 2025, 2:49 AM

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#2 h Mar 16, 2019, 8:53 AM • 1 Y

Y by Adventure10

$bump$ ...

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#3 h Mar 16, 2019, 9:25 AM • 2 Y

Y by Adventure10, Mango247

1. Simple:Place the earthlings first. $\implies m!$ ways.
Now, there are $m+1$ gaps. Hence, place n martians in that . $\implies \binom{n}{m+1}$

So, $m! \binom{n}{m+1}$ ways. Note that $n>m+1$

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#4 h Mar 16, 2019, 9:26 AM • 2 Y

Y by Adventure10, Mango247

For a round table. $m! \to (m-1)!$
$\binom{n}{m+1} \to \binom{n}{m}$
So, $(m-1)! .\binom{n}{m}$ ways

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#5 h Mar 17, 2019, 3:46 AM • 2 Y

Y by Adventure10, Mango247

chem123 wrote:

1. Simple:Place the earthlings first. $\implies m!$ ways.
Now, there are $m+1$ gaps. Hence, place n martians in that . $\implies \binom{n}{m+1}$

So, $m! \binom{n}{m+1}$ ways. Note that $n>m+1$

in the first line ,
why $m!$ ways,earthlings are $n$ numbered.so wouldn't that be $n!$ ?

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#6 h Mar 17, 2019, 6:55 AM • 1 Y

Y by Adventure10

why not $2$ , will be $(m-1)! \times n!$ ?

you may seee: https://artofproblemsolving.com/community/c3h1804185_boys_and_girls

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#7 h Mar 17, 2019, 7:19 AM • 2 Y

Y by Adventure10, Mango247

Sorry. In all my answers, interchange $m$ and $n$ .

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Richie

#8 h Mar 17, 2019, 11:36 AM • 1 Y

Y by Adventure10

a)
b)

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#9 h Mar 17, 2019, 12:14 PM • 2 Y

Y by Adventure10, Mango247

Richie wrote:

a)
b)

can you please explain? that will be really helpful

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#10 h Mar 18, 2019, 12:18 PM • 2 Y

Y by Adventure10, Mango247

how is there are always $n$ and $n+1$ gaps between earthligs????? Please help

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#11 h Mar 18, 2019, 12:32 PM • 2 Y

Y by Adventure10, Mango247

S(a)

S(b)

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#12 h Mar 18, 2019, 1:37 PM • 1 Y

Y by Adventure10

So, there are $m+1$ places for seating the earthlings with at least one in each place and so this can be done in $\binom{n-1}{m}$

don't understand,can u please explain deeply

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Richie

#13 h Mar 18, 2019, 6:41 PM • 4 Y

Y by anonman, third_one_is_jerk, Adventure10, Mango247

Hate to say it, but Purple_Planet 's answer and solution are wrong. We should first place the earthlings not the martians. Here is my solution.
S(a)
S(b)

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#14 h Mar 19, 2019, 7:41 AM • 2 Y

Y by Adventure10, Mango247

I understand everything except in $a$ you said that then there are $n+1$ gaps in which we have to place the martians.

But will there always be $n+1$ gaps ? like in the picture, there are $n-1$ gaps placing $2$ earthlings at the start and end of the line.what do you think?

OMG,it's $500$ th post!!!

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Richie

#15 h Mar 19, 2019, 7:50 AM • 2 Y

Y by Adventure10, Mango247

The Martians can also sit to the left of the first Earthling and also to the right of the last Earthling like this below:- $M E M E E M$

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#16 h Mar 19, 2019, 7:53 AM • 2 Y

Y by Adventure10, Mango247

Richie wrote:

The Martians can also sit to the left of the first Earthling and also to the right of the last Earthling like this below:- $M E M E E M$

but wasn't that meant to be the end and starting point of the line? then how can you place martinas left of the first earthling and to right of the last earthling?

then wouldn't it go outside the line?

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Richie

#17 h Mar 19, 2019, 7:58 AM • 3 Y

Y by anonman, Adventure10, Mango247

There is nothing mentioned about whether the line must start and end with an Earthling. So we can put the Martians at the two extreme ends as well.

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#18 h Mar 19, 2019, 7:59 AM • 2 Y

Y by Adventure10, Mango247

okay,thank you very much!

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