I got the children's one wrong but, on reflection, I am a little confused. I agree that children, being the plural, would have the possesive adjective between the n and the s. For instance the children's clothes would indicate the clothes belonging to the children. What about a children's clothes department though, as in the department selling clothes that fit children? The apostrophe would be wrong in this case, would it not?

My brain hurts...

:D (tG)