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

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;
d) there are 5 men and they must sit next to each other;
e) there are 4 married couples and each couple must sittogether.

Answer

a) there are 8! = 40320 ways.

b) There are 7 ways to choose 2 adjacent seats for personsAand 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.
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
4! * 4! * 2 = 1152 ways.

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