Question

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 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?

Short answer

Part (a) if there are no restrictions on the seating arrangement, then the arrangement is 40320

Part(b) If persons A and B must sit next to each other , then the arrangement will be 10080

Part(c)there are 4 men and 4 women and no 2 men or 2 women can sit next to each other, then the arrangements will be 1152

Part(d)there are 5 men and they must sit next to one another. then the arrangement will be 2880

Part(e)there are 4 married couples and each couple must sit together, then arrangements will be 384

Step-by-step solution

Part (a) given information 

here We have to find out the arrangements among 8 people if there are no restrictions on the seating arrangement

.Part(a).  there are no restrictions on the seating arrangement 

if there no restrictions then the permutation will be$8!=40320$

    Step 1 Part (b). Given information

We have to find the seating arrangements among 8 people if persons A and B must sit next to each other

Step 2. Part(b) .persons A and B must sit next to each other 

If A and B sit next to each other they will form a group, then there are 7 objects and the arrangement between A and B is 2!. Thus the seating arrangement will be$7!\times 2!=10080$

    Step 1 Part (c). Given information

We have to find the seating arrangements among 8 people if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other

Step 2. Part(c) .  there are 4 men and 4 women and no 2 men or 2 women can sit next to each other 

If 4 men and 4 women each are in a group, then there will form 2 groups and the permutation of this group is 2! and within a group of men and women arrangement will be $4!\times 4!$. Thus if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other , then the arrangement will be$4!\times 4!\times 2!=1152$

Part (d) given information  

here We have to find out the arrangements among 8 people if there are 5 men and they must sit next to one another

.Part(d). there are 5 men and they must sit next to one another 

if 5 men and they must sit next to one another then this will form a group and arrangements within the group will be 5!.So there will be one men group and 3 women. thus the permutation will be$5!\times 4!=2880$

Part (e) .Given information  

We have to find out the arrangements among 8 people if there are 4 married couples and each couple must sit together

.Part(e). there are 4 married couples and each couple must sit together 

If there are 4 married couples and each couple must sit together, there will form 4 pairs of couples and the permutation will be 4! The arrangements within each couple is 2! and so permutation of the given question will be$4!\times 2!\times 2!\times 2!\times 2!=384$