Find circular arrangements

Question

Find the possible ways to arrange 3 persons in 5 places in a circle in each of the following:

If it is not necessary to the persons to be neighboring
If it necessary to be neighboring

My answer:

$_{n-1}P_{r-1} = {}_4P_2$
$nr! = 5\cdot3!$

But how can the number of ways in the second case be larger than the number of ways in the first case even though the first case includes the second case?

score 0 · Answer 1 · answered Oct 13 '17 at 17:11

1) $\frac {1}{5} 5\cdot 4\cdot 3$

There are 5 place for Alice to sit, 4 places of Bob to sit 3 places for Carol to sit, and $\frac {1}{5}$ for the circular table.

2) $3!$

Must sit together, then we can fix the point between the two empty chairs, and now we have the problem of three people in a line.

score 0 · Answer 2 · answered Oct 13 '17 at 17:13

0

In your first answer, $_4P_2$ assumes that arrangements are indistinguishable if they are rotationally symmetric. However, the 5 in your second answer implies that they are distinguishable; by multiplying 5, you are counting the arrangement ABC 5 times, once for each location around the circle.

answered Oct 13 '17 at 17:13

louisng114

13

Does that mean the second answer is not correct ? – Hussien Mohamed Oct 13 '17 at 18:48
Yes; if they are neighboring in a circle, under the same condition as the first question where only relative location matters, $5 \cdot 3!$ is not the correct answer. – louisng114 Oct 14 '17 at 01:52

Find circular arrangements

2 Answers2