Q.1.10 - A First Course in Probability

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? 

Step-by-Step Solution

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 1 Part (a) given information

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

Step 2 .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 $4!\times 4!\times 2!=1152$

Step 1 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 

Step 2 .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 $5!\times 4!=2880$

Step 1 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 

Step 2 .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 $4!\times 2!\times 2!\times 2!\times 2!=384$