In how many ways can 5 people be seated on 7 chairs a round table if
a) only their positions relative to each other count (that is, the arrangements obtained from each other by rotation of all people are considered to be the same)
First we will calculate the number of ways the 5 could be seated in the 7
chairs if the chairs were in a row.
Then we will divide by 5 because for every way they could be seated at a round
table, there are 5 ways they could sit in the row of chairs that would amount
to the same seating arrangement at the round table.
So first we work the problem as if the chairs were in a row:
If they were in a row. there would be
7 places to seat person #1, leaving
6 places to seat person #2, leaving
5 places to seat person #3, leaving
4 places to seat person #4, leaving
3 places to seat person #5.
That's 7·6·5·4·3 = 2520
Then we divide that by 5 rotations and get
7·6·4·3 = 504 ways.
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b) only who sits next to whom counts, but not on which side (rotations and reflections do not change the arrangement)?
Each seating arrangement in part (a) has one reflection, so we divide that answer
by 2 and get 252.
Edwin