If we now consider any one of the combinations of the boys and girls :
BBGGGBB
BGBGBGB
GBBGBBG
Let's take the first one for example:
B B G G G B B
We have 4 boys and 3 girls, so we'd have
= 4 x 3 x ( 3 x 2 x 1 ) x 2 x 1
= 4 x 3 x 3! x 2 x 1
= 4! x 3! ways (or if you want 4P4 x 3P3)
Now the other 2 combinations give the same number of ways
Let's take a second one
B G B G B G B
We'd have
4 x 3 x 3 x 2 x 2 x 1 x 1
which is the same as 4! x 3!
As we have 3 combinations
BBGGGBB or BGBGBGB or GBBGBBG
We'd have 4! x 3! + 4! x 3! + 4! x 3!
= 3 x (4! × 3!)
= 432 ways