In how many ways can 4 married couples (total of 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 sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?

Solution:

Given, there are 4 married couples

Total number of people = 8

we have to find how many ways can 4 married couples be seated in a row

a) if there are no restrictions on seating arrangement

The number of possible ways = n!

Here, n! = 8!

= 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

= 40320 ways.

Therefore, there are 40320 ways that the people can be seated when there is no restriction on the seating arrangement.

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

The possible ways = 7! × 2!

= (7 × 6 × 5 × 4 × 3 × 2 × 1) × (2 × 1)

= 5040 × 2

= 10080

Therefore, if persons A and B must sit next to each other there are 10080 ways of seating arrangement.

c) if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other

This implies the restriction of having persons of opposite sex next to each other.

We can have any one of the 8 persons in the first position.

So, both man and woman can be arranged without having same next to

each other = 8 × 4 × 3 × 3 × 2 × 2 × 1 × 1

= 1152 ways

Therefore, if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other, there are 1152 ways of seating arrangement.

d) if there are 5 men and they must sit next to one another

There are 5 men and 3 women. 5 men sit next to one another.

The possible number of ways = 5! × 4!

= (5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1)

= 120 × 24

= 2880 ways

Therefore, if there are 5 men and they must sit next to one another, there are 2880 ways of seating arrangement.

e) if there are 4 married couples and each couple must sit together

4 pairs of couples can be arranged in 4!

Each couple can be arranged in 2! Ways.

The possible number of ways = 2! × 2! × 2! × 2!

= (2× 1) × (2 × 1) × (2 × 1) × (2 × 1) × (4 × 3 × 2 × 1)

= 2 × 2 × 2 × 2 × 24

= 16 × 24

= 384 ways

Therefore, if there are 4 married couples sitting together, there are 384 ways of seating arrangement.

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In how many ways can 4 married couples (total of 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 sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?

Summary:

The possible number of ways can 4 married couples (total of 8 people) be seated in a row if:

(a) there are no restrictions on the seating arrangement is 40320 ways

(b) persons A and B must sit next to each other is 10080 ways

(c) There are 4 men and 4 women and no 2 men or 2 women can sit next to each other in 1152 ways.

(d) there are 5 men and they must sit next to one another is 2880 ways

(e) there are 4 married couples and each couple must sit together in 384 ways.