In how many ways can 2 boys and 3 girls sit in a row so that no two girls sit side by side 2?
Answers
You're overcounting the number of ways "at least two" girls can sit together, because for example
Boy1 Boy2 Girl1 Girl2 Girl3 Boy3 Boy4 Boy5
is produced twice: Once considering Girl1 Girl2 as the unit, and once considering Girl2 Girl3 as the unit. This means that you're subtracting too large a number from 8!, leading to a too small final result.
What you're doing is an inclusion-exclusion count, and for those you generally need to keep adding correction terms with alternating signs for ever more special cases.
An easier way to count is to start by finding all the allowed "gender sequences", without respect to which boys and girls are where. Afterwards you can multiply by 5!3!do distribute the actual children on the chairs.
To count the gender sequence, start by considering the three girl positions. The two first of them need to have at least one boy immediately to the right, so we have
... G B ... G B ... G ...
Now we have to distribute the three remaining B positions in among the four parts noted .... That's a stars-and-bars problem with (3+4−13)possible outcomes.
So the total count you're looking for is(63)⋅5!⋅3!