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

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