(4) In how many ways 7 persons stand in a row, if  three particular persons do not stand together  ?

Soln. 7 persons can stand in a row in 7 !  =  5040   ways                            –––––––––  ( 1 )

Three particular person stand in a row in  3 !  =  6  ways.

If  the group of  three particular person are considered as 1 then remaining 4 means total 5 persons can stand in a row in  5 ! =  120  ways.

 Total arrangement of  7 persons stand in a row ( with three particular persons stand together in a row )  =  6  120  =  720                                      –––––––––  ( 2 )

  From equation (1) & (2),  required arrangements  =  5040  –  720  =   4320

(5) In how many ways 4 men and 4 women stand in a row such that no two women stand together  ?

Soln. __ M __ M __ M __ M __

By above diagramme, 4 men can stand in 4 ! =  24 ways

And 4 women can stand in any 4 places from 5 blank spaces in    =  120  ways

  Total no. of  required arrangements  =  24  120  =  2880.

(6) In how many ways 3 men and 3 women stand in  a row such that no two men or women stand together  ?

Soln. __ M __ M __ M            or            M __ M __ M __

 For above diagramme,

Men can stand in a row in  3 !  =  6  ways

And women in    =  6  ways.

Hence, total no. of  arrangements

=  6  6   =   36  ways. For above diagramme,

Men can stand in a row in 3 !  =  6  ways

And women in   =  6  ways.

Hence, total no. of  arrangements

=  6  6   =   36  ways

Thus,  total number of  required arrangements  =  36 + 36  =  72

Best  of   Luck