Question

The number of ways of seating three gentlemen
and three ladies in a row, such that each
gentlemen is adjacent to atleast one lady, is
(1) 360
(2) 72
(3) 720
(4) None of these​

Answer 1

Answer:

Step-by-step explanation:

See we will go by your method using complements. See total ways are 720 . now lets group three men so let us assume them x . now let us place them in positions 123 so total ways are 3!.3!=36 as men and women can e arranged amongst in 3! ways. Now place x at place 234 again same number of ways ie 36 now similarly at 345,456 so total ways become 36⋅4=144 now group any two men . this can be done in 3 ways. Place them at place 1,2 now total ways become 3.2!.3.3!=108 . men can be arranged in 2! ways 3rd place has to be female or it becomez similar to 3 men together so 3 ways and 1M,2w can be arranged in 3! ways. Note that now they cant be placed in positions (23),(34),(45) as it becomes similar to 3 men together. So now they can be placed at 56 position. Again similar to 12 position we get 108 ways . now remember we are talking complement of total permitted ways . hence total ways are 720−144−216=360