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