Permutations and Combinations in Circular Arrangement

Question

8 persons sit at a round table with 10 seats so that there is exactly one person between the two empty seats. How many possible arrangements are there?

Here's what I have so far:

${10 \choose 1}$ (for choosing the seat of the person to be isolated)

(8-1)! (to permute the group of 3 + remaining 7 people around the round table)

So my solution is 10*7! number of possible arrangements. Is this correct?

Why are you considering only the "remaining 7" and not all 8 people? What do you mean by "the group of 3"? — 79037662, Sep 30 '19 at 14:03

score 3 · Accepted Answer · answered Sep 30 '19 at 14:10

It doesn't matter how the "seat between the empty seats" is chosen, because we are considering a round table. We simply need to choose one person to sit here, and arrange the remaining 7. The number of possible arrangements is thus:

$$8 \cdot 7! = 8!$$

drhab · Answer 2 · 2019-09-30T14:32:25.660

2

If the seats are distinguishable then the answer is: $10\cdot8!$.

First choose the isolated seat (factor $10$). This move also determines which $8$ seats are available.

Place the $8$ persons on the available seats (factor $8!$).

If the seats are not distinguishable then the answer is: $8!$.

Factor $10$ falls out.

edited Sep 30 '19 at 14:32

answered Sep 30 '19 at 14:20

drhab

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Permutations and Combinations in Circular Arrangement

2 Answers2