Answer:
a ) P₈  =  40320
b) P = 576
c) P =   1440
d) P =  24
Step-by-step explanation:
a )  8 people sitting in a row without restrictions
simple  Total ways  =  P₈  =  8!
P₈  =  8*7*6*5*4*3*2*1
P₈  =  40320
b) There are 4 men and 4 women  and no two men or 2 women can sit next to each other
Let  letters e women  and numbers  be men we have something like this
1  a  2  b  3  c  4  d
So we have the permutations of the four digits (men)
P₄  =  4!  
P₄  =  4*3*2*1   =  24 
And the permutations of the 4 women too  equal to 24
Then total ways of sitting in a row in this case is
P = 24*24  = 576
c) There are 5 men and they have to sit next to each other
In this case we have 2 groups:  
Group 1    5 men                        Group  2    3  women
We have two groups and the ways are
group men first                group of women second         1  way
group of women first       group of  men second             2  way
Permutations within men group
P₅  =  5!    =  5*4*3*2*1     
P₅  =   120
Permutations within the women group
P₃  = 3!     =  3*2*1  
P₃  = 6
Total ways case c)   T = 6*120*2  = 
P(c)  =  1440
d) There are four couples and each couple must be together
P₄ =  4!  =  4*3*2*1
P₄ =  24