I'm sure carrot being 6 letters long has nothing to do with it. The questions are meant to get you thinking of ANYTHING but vegetables, they all come out to 6 so you think there is some trick involved.. ie- is it really 6? double-check.. okay.. next..
Then they come out of left field and ask for a vegetable. Carrot is common and easy to think of first, thats all there is to it.