The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of 7 chairs such that at least 2 boys are next to each other?
Again this one is easier if we consider the distributions where 2 boys are not next to each other and subtract that off of the total number.
The only distribution that has no boys together is B G B G B G B
The only degrees of freedom we have is in selecting which girls and which boys get which seat.
There are 4! ways to seat the boys, and 3! ways to seat the girls that gets us N=4! 3! = 144
There are a total of 7!=5040 possible ways to seat everyone
Thus the number of seatings where at least 2 boys are adjacent is 5040-144 = 4896