Hello, sweeetcaroline!

I found no elegant approach to this problem.

I was forced to use "brute force" listing . . . *blush*

Suppose the colors are Red, Blue, and Green.3 pairs of socks are placed side-by-side of a straight clothes line.

The socks in each pair are identical, but the pairs themselves are of different color.

If I stand on one side of the line, how many different color arrangements can I see

if no sock is next to its mate?

Then we have: .

We have 3 choices for the color of the first sock (#1).

Suppose it is Red.

The other Red can be in positions #3, 4, 5 or 6.

Reds in (1,3): .

Then there are two cases:

. . #2 is , and the last three must be

. . #2 is , and the last three must be

Reds in (1,4): .

Then there are four cases:

. . #2, 3 are , and the last two are or

. . #2, 3 are , and the last two are or

Reds in (1,5): .

Then there are two cases:

. . #2, 3, 4 are , and the last is

. . #2, 3, 4 are , and the last is

Reds in (1,6): .

Then there are two cases:

. . #2, 3, 4, 5 are

. . #2, 3, 4, 5 are

Hence, there are: . arrangements that begin with Red.

Since there are 3 choices for the color of sock #1,

. . there are: . arrangements with no adjacent mates.