Question

Four married couples have purchased 8 seats in a row for a movie. in how many different ways can they be seated if:

a.there are no restrictions on seating?

b.if each couple must sit together?

c.if all men sit together to the right of all women?

d.if the seating arrangement must alternate genders, that is bgbgbgbg or gbgbgbgb?

e.if the seating arrangement must alternate genders, and each couple must sit together?

Answer 1

Answer:

a. There are no restrictions on seating. Then there are a total of 8 people that can be seated in 8 seats.

So, total number of ways is $8! = 40,320$

b. If each couple must sit together. Then there are only 4 pairs to be seated.

It means 4 pairs can be arranged in 4! ways.

But because the pairs can also interchange there positions we have $2^{4}$ shufflings possible.

Therefore, total number of ways is $4!* 2^{4}= 384$

c. If all men sit together to the right of all women. There are 4 men and 4 women. Each can be seated in 4! ways.

Thus, total number of arrangement possible is $4! * 4! = 576$

d. If the seating arrangement must alternate genders, that is bgbgbgbg or gbgbgbgb.

Total number of arrangement possible $= 8 * 4 * 3 * 3 * 2 * 2 * 1 * 1 = 1152$

e. if the seating arrangement must alternate genders, and each couple must sit together.

Total number of arrangements = $2 * 4! = 48$