What is the correct answer to the distinct equivalence classes of R?

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

Strings

2, 3 zeroes 4

1,1 2 zeroes 6

1, 3 zeroes 4

all zeroes 1

So 15 possible strings

An equivalence class is a partition of these strings into subsets

The number of such partitions is the Bell number with n=15

You can look this up