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?

Answer: (A) 40320 ways(B) 10080 ways(C)1152 ways(D) 2880 ways(E) 384 waysStep-by-step explanation:(A) if there are no restrictions then all 8 people can be arranged in = 8! = 40,320 waysB) if A and B sit together then they made a “block” which can be arranged along with the remaining 6 people in 7! ways. But the A and B can also be arranged in 2! ways inside the block.Therefore,Ways of arranging 8 people when two people(A and B) must be together = 2!*7! = 10,080 ways C) the restrictions impose having persons of opposite sex next to each other . this can be done this wayOn the first position we can have any of the 8 persons from either man or woman.for one choice of the person for the second position we shall have 4 possible choices for the second position among the people of opposite sex , then for any particular choice of the first two positions we have 3 choices for the third position among the remaining people of the same sex with the person on the first position(and consequently of different sex with the one on the second position) and so on .. So both man and woman can be arranged without having same next to each other = 8*4*3*3*2*2*1*1 ways = 1152 ways D) the men can be considered a “block” and permuted along with the remaining 3 women in 4! ways and as we can also permute the men within the block in 5! ways then the total amounts to 5!*4!=2,880 ways E) Since there are 4 couples which make 4 blocks.4 blocks can be arranged in 4! waysSince each couple can arranged 2! ways in each of the blocks making 2!*2!*2!*2! ways = 16ways for all block. Therefore, the arrangement will give 16*4!= 384 ways