Justin T. answered • 11/14/14
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Let the 5 boys be a,b,c,d,e and the 3 girls be 1,2,3. Then one arrangement that satisfies the conditions is
1,a,2,b,3,c,d,e. And some other arrangements are:
1,a,2,b,3,c,e,d
1,a,2,b,3,d,c,e
1,a,2,b,3,d,e,c
...
Since order is important, any particular arrangement of girls has 5 boys to arrange, and the number of ways to arrange 5 distinct* elements in an ordered list is just 5! or for any particular arrangement of girls there are 5! ways to arrange the boys, so the total number of arrangements then, N = (total number of distinct arrangements of girls, G)(5!).
In considering what makes one arrangement of girls distinct from another, we would have to consider the order of the girls relative to each other and the physical positions in the row the girls are in (i.e. 1,a,2,b,3,c,d,e is distinct from 1,a,b,2,c,3,d,e is distinct from 1,a,3,b,2,c,d,e). All possible relative orderings of the girls are:
1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1
or 3!. Since for anyone of these particular orderings the girls must be at least 1 space apart, the number of arrangements any particular ordering could have would depend on the positions of the girls relative to each other and would only have to consider the first girl up to the 4th position in the row, since for any arrangement where the first girl is past the 4th position the order under consideration would no longer hold, or for the ordering (1,2,3):
1,a,2,b,3,c,d,e. And some other arrangements are:
1,a,2,b,3,c,e,d
1,a,2,b,3,d,c,e
1,a,2,b,3,d,e,c
...
Since order is important, any particular arrangement of girls has 5 boys to arrange, and the number of ways to arrange 5 distinct* elements in an ordered list is just 5! or for any particular arrangement of girls there are 5! ways to arrange the boys, so the total number of arrangements then, N = (total number of distinct arrangements of girls, G)(5!).
In considering what makes one arrangement of girls distinct from another, we would have to consider the order of the girls relative to each other and the physical positions in the row the girls are in (i.e. 1,a,2,b,3,c,d,e is distinct from 1,a,b,2,c,3,d,e is distinct from 1,a,3,b,2,c,d,e). All possible relative orderings of the girls are:
1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1
or 3!. Since for anyone of these particular orderings the girls must be at least 1 space apart, the number of arrangements any particular ordering could have would depend on the positions of the girls relative to each other and would only have to consider the first girl up to the 4th position in the row, since for any arrangement where the first girl is past the 4th position the order under consideration would no longer hold, or for the ordering (1,2,3):
If girl 1 is in the row's 1st position and:
girls 2 & 3 are 1 space apart, then there are 4 such arrangements.
2 & 3 are 2 spaces apart, then 3 arrangements
3 spaces apart, 2 arrangements and 4 spaces apart, 1 arrangement
or 4+3+2+1=10 arrangements with a girl in the 1st position. If girl 1 is in row's 2nd position and:
girls 2 & 3 are 1 space apart, then there are 3 such arrangements.
2 & 3 are 2 spaces apart, then 2 arrangements
3 spaces apart, 1 arrangement
or 3+2+1=6. Continuing this process, for the 3rd position, 3 arrangements, and 4th position, 1 arrangement, so for any particular ordering of girls there are 10+6+3+1 ways to place them in the row. So the total number of distinct arrangements of girls G = (3!)(10+6+3+1)=120, so then N = 120(5!) = 14,400
girls 2 & 3 are 1 space apart, then there are 4 such arrangements.
2 & 3 are 2 spaces apart, then 3 arrangements
3 spaces apart, 2 arrangements and 4 spaces apart, 1 arrangement
or 4+3+2+1=10 arrangements with a girl in the 1st position. If girl 1 is in row's 2nd position and:
girls 2 & 3 are 1 space apart, then there are 3 such arrangements.
2 & 3 are 2 spaces apart, then 2 arrangements
3 spaces apart, 1 arrangement
or 3+2+1=6. Continuing this process, for the 3rd position, 3 arrangements, and 4th position, 1 arrangement, so for any particular ordering of girls there are 10+6+3+1 ways to place them in the row. So the total number of distinct arrangements of girls G = (3!)(10+6+3+1)=120, so then N = 120(5!) = 14,400
*assuming human beings are distinct and thus, each boy and girl can be considered distinct "elements"
