Q: 8 boys and 2 girls sit on the bench. if the girls may sit neither at the ends nor together. In how many ways can they be arranged?
8 boys and 2 girls = 10 total
with no restriction 10!
neither at the ends nor together thus the events
a) not at the ends
for this part i believe that the two girls can sit next to each other
so there are 10 spaces on the bench
_ _ _ _ _ _ _ _ _ _
the two spaces at the ends are for the boys so
it is
B _ _ _ _ _ _ _ _ B
8 7
there are 6 boys & 2 girls left 8 in total; there are 6! ways of ordering the boys and 2! ways of ordering the girls
= 8 x 7 x 2 ! x 6! = 80640
b) not together
g g b b b b b b b b b
there are 2! ways of arranging the girls and 9! ways of arranging them all
so it will be 10! - (2! x 9!) = ways
since the events are mutually exclusive thus
80640 + 10! - (2! x 9!) = 2983680 ways
I have tried several times but i cannot get the answer 1693440
Please can you help