In how many ways can 4 boys and 4 girls be seated in a row if
a. they can seat anywhere b.
There are 8 people.
We can choose the person to sit in the 1st chair 8 ways.
That leaves 7 people to seat and 7 vacant chairs.
We can choose the person to sit in the 2nd chair 7 ways.
That leaves 6 people to seat and 6 vacant chairs.
We can choose the person to sit in the 3rd chair 6 ways.
That leaves 5 people to seat and 5 vacant chairs.
We can choose the person to sit in the 4th chair 5 ways.
That leaves 4 people to seat and 4 vacant chairs.
We can choose the person to sit in the 5th chair 4 ways.
That leaves 3 people to seat and 3 vacant chairs.
We can choose the person to sit in the 6th chair 3 ways.
That leaves 2 people to seat and 2 vacant chairs.
We can choose the person to sit in the 7th chair 2 ways.
That leaves 1 person to seat and 1 vacant chair.
We can only seat this person in the 8th chair.
Answer = 8∙7∙6∙5∙4∙3∙2∙1 = 8! = 40320
the boys and the girls are to be seated alternately?
Case 1. The boys sit in seats 1,3,5,7 and the girls sit in seats 2,4,6,8.
We can choose the boy to sit in the 1st chair 4 ways.
That leaves 3 boys to seat and 3 vacant chairs for boys.
We can choose the boy to sit in the 3rd chair 3 ways.
That leaves 2 boys to seat and 2 vacant chairs for boys.
We can choose the boy to sit in the 5th chair 2 ways.
That leaves 1 boy to seat and 1 vacant chair.
We can only seat this boy in the 7th chair.
We can choose the girl to sit in the 2nd chair 4 ways.
That leaves 3 girls to seat and 3 vacant chairs for girls.
We can choose the girl to sit in the 4th chair 3 ways.
That leaves 2 girls to seat and 2 vacant chairs for girls.
We can choose the girl to sit in the 6th chair 2 ways.
That leaves 1 girl to seat and 1 vacant chair.
We can only seat this girl in the 8th chair.
That's 4∙3∙2∙1∙4∙3∙2∙1 = 4!4! for Case 1.
Case 2. The boys sit in seats 2,4,6,8 and the girls sit in seats 1,3,5,7.
That's the same answer as Case 1, 4!4!
Final answer: 4!4!+4!4! = 2(4!)2 =2(24)2 = 1152
Edwin
