Question

Seating Arrangement for 4 Boys and Girls
The number of which 4 men and 4 women can be seated at a round table so that no women sit together. In how many ways can 4 boys and 4 girls be arranged in a row such that no two boys and no two girls are next to each other? (A) 1032 (B) 1152 (C) 1254 (D) 1432 (E) 1564

Answer 1

The following answers do not match the given numbers... perhaps the given options are not correct...

1) Around a table:

The number of ways of arranging 4 boys around in a circle:  4! /2 = 12. We divide by 2 because of arrangement clockwise or anticlockwise is considered same.

There will be 4 gaps between girls. So the number of ways of arranging them is  = 4! = 24
      So the answer = 12 * 24 = 288

2)  In a row:

 (a)  The number of ways of arranging 4 boys in a line =  4! = 24
   Then there are 5 locations where girls can be seated, at the ends or in between boys.
       So the number of ways of arranging women = 5P4 = 5! = 120
                                            = 120 * 24 = 2, 880

  (b) Suppose two boys are always together without a woman in between. There are still 4 gaps available for 4 women, so that they do not sit together.  Two men in 4P2, then third man can be selected in 2 ways. Then 4 women in 4P4 ways.

    So the number of ways :  4P2 * 2 * 1 * 4P4 = 12 * 2 * 1 * 4!  = 576

  So finally, the total number = 2,880 + 576 = 3,456

Answer 2

1) Around a table: The number of ways of arranging 4 boys around in a circle:  4! /2 = 12. We divide by 2 because of arrangement clockwise or anticlockwise is considered same. There will be 4 gaps between girls. So the number of ways of arranging them is  = 4! = 24. So the answer = 12 * 24 = 288

2)  In a row: (a)  The number of ways of arranging 4 boys in a line =  4! = 24. Then there are 5 locations where girls can be seated, at the ends or in between boys.  So the number of ways of arranging women = 5P4 = 5! =   = 120 * 24 = 2, 880

(b) Suppose two boys are always together without a woman in between. There are still 4 gaps available for 4 women, so that they do not sit together.  Two men in 4P2, then third man can be selected in 2 ways. Then 4 women in 4P4 ways.

 So the number of ways :  4P2 * 2 * 1 * 4P4 = 12 * 2 * 1 * 4!  = 576

So finally, the total number = 2,880 + 576 = 3,456