a) The only he gets one point is with 6 on 1st dice and 2,3 or 4 on the 2nd dice or the other way round.

First probability is 1/6x3/6=1/12 Second probabilty 3/6x1/6=1/12 So answer = 2/12 = 1/6

This is effectively the same as you did so I suspect the book is wrong.