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 each other? (e) there are 4 married couples and each couple must sit together?

a ) P(8)   =  40320  

b ) Pt(7) =  10080

c ) P(t)   =  48

d )  P(4)  =  24

e ) P(4)  =  24

Step-by-step explanation:

a) Without restrictions  8 persons can seat according to

P(8)   =  8 !          P(8)   = 8*7*6*5*4*3*2*1

P(8)   =  40320

b) Two people must sit next to each other

Let   A  and   B are a unit in the group

Then we have 7 elements

P₁(7)   =  7 !           P₁(7)   =  7*6*5*4*3*2*1

P₁(7)  = 5040

Now this number is for arrangement  A  B   we should add the same number now with  B  A

Then  P₂(7)  = 5040  and

Pt(7) =  10080

c)  There are 4 men and 4 women that can not sit next to each other

Let   A B C  and D      stands for men   and  1  2  3  and  4  women

A 1 B 2 C 3 D 4

This kind of arrangement is what we are loking for

If we change men and women are still we have

P₁(4)   =  4!               P₁(4)  =  4*3*2*1

P₁(4) = 24

Now we dont move men but women

P₂(4) = 24

In this case P(t)   =  24+24

P(t)   =  48

d) If five men is a group we have only 4 elements

P(4)  =  4!         P(4)  = 4*3*2*1

P(4)  =  24

e) If we have four couples we only have four elements

P(4)  =  24