# Possibilities to place n persons on a round table with $2n$ chairs so that there is at least one free chair between each $2$ persons

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?

$$2n$$ 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.

• "possessed" Halloween, eh? – Parcly Taxel Oct 31 '18 at 9:56
• @ParclyTaxel Oeps, my English. Anything wrong with "possessed"? Any link with Halloween? – drhab Oct 31 '18 at 9:58
• Nothing wrong, except that it appears to be a link. – Parcly Taxel Oct 31 '18 at 9:59
• @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. – Barry Cipra Oct 31 '18 at 10:14
• @BarryCipra I see. In Dutch language we have "bezet" for the first (harmless) meaning and "bezeten" for the second. – drhab Oct 31 '18 at 10:22