0
$\begingroup$

How many possibilities are there to place n persons on a round table with $2n$ chairs, so that there is at least one free chair between each $2$ persons?

Assuming we have $n = 10$ persons that want to sit there. So we have $n = 20$ chairs. The first person has $20$ choices. After the first person is placed onto his seat, we have $17$ choices, because the seats next to the first person can't be chosen...

So can I say the number of possibilities is $$\sum_{n=0}^n n*(n-3)$$

Or is that totally false?

And how many possibilities are there to place $n$ persons on a round table with 2n+1 chairs, so that there is at least one free chair between each $2$ persons?

$\endgroup$
2
$\begingroup$

$2n$ chairs.

There are $2$ choices for the chairs that will be possessed by persons.

After that choice there are $n!$ possible arrangements for the persons, so there are $2\cdot n!$ possibilities.


$2n+1$ chairs.

There will be exactly $2$ consecutive chairs that will not be possessed and selecting them comes to the same as selecting the chairs that will be possessed. This gives $2n+1$ choices.

After that choice there are $n!$ possible arrangements for the persons, so there are $(2n+1)\cdot n!$ possibilities.

$\endgroup$
  • 1
    $\begingroup$ "possessed" Halloween, eh? $\endgroup$ – Parcly Taxel Oct 31 '18 at 9:56
  • $\begingroup$ @ParclyTaxel Oeps, my English. Anything wrong with "possessed"? Any link with Halloween? $\endgroup$ – drhab Oct 31 '18 at 9:58
  • $\begingroup$ Nothing wrong, except that it appears to be a link. $\endgroup$ – Parcly Taxel Oct 31 '18 at 9:59
  • $\begingroup$ @drhab, "possessed" as a (past tense) verb means "occupied." As an adjective, it means "controlled by an indwelling demon," as in The Exorcist. Hence the reference to Halloween. $\endgroup$ – Barry Cipra Oct 31 '18 at 10:14
  • $\begingroup$ @BarryCipra I see. In Dutch language we have "bezet" for the first (harmless) meaning and "bezeten" for the second. $\endgroup$ – drhab Oct 31 '18 at 10:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.