In how many ways can 8 people be seated in a row if

a) there are no restrictions on the seating arrangement;

b) persons A and B must sit next to each other;

c) there are 4 men and 4 women and no 2 men or 2 women can

sitnext to each other;

sitnext to each other;

d) there are 5 men and they must sit next to each other;

e) there are 4 married couples and each couple must

sittogether.

sittogether.

## Answer

a) there are 8! = 40320 ways.

b) There are 7 ways to choose 2 adjacent seats for personsAand

B.

B.

and there are 2 ways to choose which seat A gets and whichseat

B gets, and there are 6! ways to seat the remaining 6people.

B gets, and there are 6! ways to seat the remaining 6people.

7 * 2 * 6! = 10080 total ways.

c) since no 2 men women can sit next to each other, we

musthave men and women alternating in the 8 seats. There are 4!

ways toarrange the 4 men,and 4! ways to arrange the 4 women, 2 way

toselect whether a man or woman gets the first seat, so

musthave men and women alternating in the 8 seats. There are 4!

ways toarrange the 4 men,and 4! ways to arrange the 4 women, 2 way

toselect whether a man or woman gets the first seat, so

4! * 4! * 2 = 1152 ways.