help counting

you cannot paint the 1 and the 4  as the two faces

there are    6 c 2   ways to paint two faces = 15 ways

    and ONE way (1 and 4) that you cannot paint    = 14 ways

You forgot 2 and 2 EP, thus, there are 13 working cases out of 15 total cases. 

I agree with EP on this problem; there is no way to paint 2 and 2 because

there is only one 2.

Those that work:

   1,2     1,3     1,4     1,5     1,6

   2,3     2,4     2,5     2,6

   3,4     3,5     3,6

   4,5     4,6

   5,6

The one that doesn't:  1,4