Question

how many ways can 7 people be arranged in a circle if two must always sit together

Answer 1

There are 7!/7 ways 7 people can sit around a table. If two of the people are fixed sitting together, it is as if there are 6 people sitting at the table (5 people plus a pair of two fixed people). There are 6!/6 ways to arrange the six people around the table, however since our pair of two people can be permuted there are really 6!/6 * 2! ways to seat them. This is equal to 240.

Answer 2

why is it 7! over 7 and not just 7!

Answer 3

@combinatricsandstuff, think about it with a simpler example of two people sitting at a table (or standing in a circle). How many different ways can two people sit around a table? It is tempting to say 2! = 2 ways (one person at twelve o'clock/north and one person at six o'clock/south), but since we must consider that rotations of the table does not affect the order of the people we divide by the total number of people at the table. Thus there are 2! / 2 = 1 way to arrange 2 people around a table.

Answer 4

See the above attachment for a nice visual. Keep asking great questions!