How many ways can four married couples sit in a row on a park bench if the men and women must alternate?

Question: How many ways can four married couples sit in a row on a park bench if the men and women must alternate?





Answer: 1152





How do I get to the answer?





Can anyone please help? Thank you.How many ways can four married couples sit in a row on a park bench if the men and women must alternate?
Eight people are seated in 8 positions, and there are 4 males and 4 females





How many people can sit on the first position? Any of the 8.





How people can sit on the second position? 4 (The person who sits in the second position must be of the opposite sex to the person who sat in the first position but can be any of the 4 members of that sex)





How many people can sit in the third position? 3 (The person who sits in the third position must be the same sex as the person who sat in the first position and as one member of this sex is already seated, there are three members of this sex left).





How many people can sit in the fourth position? 3 (The person who sits in the fourth position must be the same sex as the person who sat in the second position and as one member of this sex is already seated, there are three members of this sex left).





How many people can sit in the fifth position? 2 (The person who sits in the fifth position must be the same sex as the person who sat in the first position and as two members of this sex are already seated, there are two members of this sex left).





How many people can sit in the sixth position? 2 (The person who sits in the sixth position must be the same sex as the person who sat in the second position and as two members of this sex are already seated, there are two members of this sex left).





How many people can sit in the seventh position? 1 (The person who sits in the seventh position must be the same sex as the person who sat in the first position and as three members of this sex are already seated, there is one memberof this sex left).





And of course, the last person must sit in the 8th position.





So the number of ways of achieving this is 8 . 4 . 3 . 3 . 2 . 2 . 1 . 1 = 1152.





In factorial form this is 8 * 4! * 3! = 1152.How many ways can four married couples sit in a row on a park bench if the men and women must alternate?
In some states the answer could be zero :-)

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