What is the correct answer to listing the distinct equivalence classes of R in set-roster notation?

Question

Let A equal the set of all strings of 0's, 1's, and 2's that have length 4 and for which the sum of the characters in the string is less than or equal to 2. Define a relation R on A as follows:

For every s, t ∈ A, s R t  ⇔  the sum of the characters of s equals the sum of the characters of t.

It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)

Answer 1

The largest sum is 8

sum of zero 0000

sum of one 0001

0010

0100

1000

sum of 2:

0002 0011 1010

0020 0101 1100

0200 1001

2000 0110

sum of 3:

0003 0021 2010 1002 1011

0030 0201 2100 1020 1110

0300 2001 0012 1200 1101

3000 0210 0102 0111

sum of 4:

0013 3100 0202

0103 0031 2002

1003 0301 2020

3001 0310 2200

3010 0022

get the idea?