Question

Five people are to dine at a rectangular table but the host cannot decide on a seating arrangement in how many ways can the guests be seated?

Answer 1

`56xx5! =6720`

Explanation:

Arbitrarily calling one side of the table the "First Side" and labeling the others "Second", "Third" and "Fourth".

There is `1` arrangement that places `5` people on the "First Side"
and `3` arrangement that place `4` people on the "First Side"
and `6` arrangements that place `3` people on the "First Side"
and `10` arrangements that place `2` people on the "First Side"
and `15` arrangements that place `1` person on the "First Side"
and `21` arrangements that place `0` people on the "First Side"
for a total of
`color(white)("XXX")56` seat arrangements

Arranging the 5 guests (starting by seating with the left-most seat on the "First Side") can be done in `5!` ways.

This gives `56xx5!` ways of seating the guests.