We're given the following problem:
"The number of arrangements where no wife is sitting next to her husband when three married couples are seated together in the cinema (occupying six consecutive seats) is n. Find n."
Here's my approach. I first counted the number of ways a husband and a wife can sit together while making sure that the reaming people do not sit as couples. So, for the sake analogy, I imagined the follwoing: $$M_1\ M_2 \ \ A_1\ A_2 \ \ B_1 \ B_2$$ where $M_1$ and $M_2$ is the couple that will sit together.
The number of ways the couple can sit together (given that $6$ seats are available) is $3 \cdot2$ because they can be together in $3$ ways, and both of them can switch places (hence the $2$). After this, we have to sit $A_1$ and $A_2$ among the remaining $4$ chairs. This can be done in $4 \cdot 2 \cdot 2$, since $A_1$ can sit in $4$ places, and $A_2$ can sit in $2$ places (since $A_2$ cannot be next to $A_1$) (and since they can switch place, we multiply by $2$ again). Then for $B_1$ and $B_2$ , there is only two sears left, which they can sit in $2$ ways. Multiplying all this together game me $192$.
Then, I counted the number of ways two couple can sit together (making sure the thrid couple would not). In a similar way, I found that it was $48$
Thus, we find that $n=192+48 = 240$ which is correct.
The problem is, now that I think about it, is that when I counted the number of ways the couple could sit together while making sure the rest of the couples do not, I did count (I think) the following: $$A_1B_1B_2A_2M_1M_2$$ where there is actually two couples sitting together (this configuration is just an example, there are few others like this one). So how come I found the correct result? (I'm asking this because my approach seems bizarre while re-reading it).
Also, we were given a hint to solve this problem using the inclusion-exclusion, and I don't know how I'm supposed to do this (which is why I tried an intuitive approach).