How many ways to line up a family? A family with two parents, two daughters, and two sons line up for a photograph. How many ways are there for the family to line up so that the mother is next to at least one of her two daughters.
Number of possibilities next to daughter 1 + Number of possibilities next to daughter 2 - Number of possibilities next to both daughters and got
$$2 \cdot 4!+ 2 \cdot 4! - 3 \cdot 3!$$ 
Is this correct?
 A: Your approach using inclusion-exclusion is good, but the counts are off. The first term is almost right, but if the mother and daughter form a block, that still leaves $5$ units to permute, not $4$, so it should be $2\cdot5!$; likewise the second term. For the third term, again $3!$ should be $4!$ since there are still $4$ units to permute (father, two sons, one mother-daughter block); and also the factor here should be $2$, not $3$, since there are only $2$ ways to place the two daughters on both sides of the mother.
A: Find the number of ways to arrange everyone but mother.  5 people -- (5!) 
The mother can be on the left side or the right side of either daughter. (4*5!).
But we have double counted the possibility that she is between her two daughters.  
There are 2*(4!) arrangements such that the daughters are next to each other (before mom was placed).  
4*5! - 2*4!  Which looks to be the same answer as joriki placed just before me, with a slightly different way of getting there.
A: Another approach is to count the number of ways the mother can be seated next to neither daughter, and subtract from $6!$.  You get the answer
$$6!-(2\cdot3\cdot4!+4\cdot3\cdot2\cdot3!)$$
That is, to keep the mother seated away from her daughters, the mother is either seated at one of the $2$ ends with one of the $3$ males next to her and the other $4$ family members seated in the remaining seats, or else she is seated in one of the $4$ interior seats, with one of the $3$ males on her left, one of the remaining $2$ males on her right, and the final $3$ family members in the remaining seats.
(Note, introducing "seats" is just for expository convenience.  You can have the family stand up at the end and remove the seats.)
