our friends met every week to play cards, using a table with five chairs. Eventually, they realized they had chosen a different seating arrangement each week and had exhausted every possibility. How long had the friends played cards together?

If the table is circular, they played cards for 24 weeks.

If the table is rectangular, they played cards for 120 weeks.

Step-by-step explanation:

Five friends met every week to play cards, using a table with five chairs. Eventually, they realized they had chosen a different seating arrangement each week and had exhausted every possibility.

It means they have used all possible way to arrange five people in a circle.

For circular table, then total ways to arrange 5 people on a circular table = (5-1) ! = 4! = 4x3x2x1 = 24 ways.

For rectangular table, then total ways = 5! = 5x4x3x2x1 = 120 ways.