List all 3 part compositions of 5

Question

I am looking at a past exam written by a student. There was a question I believed he got correct but received only 1/4. The marker wrote down "4 more compositions, order matters".

This is the problem:

List all 3 part compositions of 5. (recall that compositions have no zeros)

$(1, 1, 3) (1, 2, 2) (1, 3, 1) (3, 1, 1) (2, 1, 2) (2, 2, 1)$ (what the student has on paper) EDIT: I made some mistakes copying

My guess is that the student wrote only (1,1,3) and (2,2,1) ,but corrected it after exam was returned. I just want to make sure that this is the case, so that I don't miss something.

Answer 1

What the student has is still wrong, assuming that you didn't typo it in writing up the question: $(3,1,1)$ has been duplicated, and $(1,2,2)$ has been omitted. Your understanding is correct, and my guess is the same as yours.

Answer 2

You're correct... the only possible ways to add 5 from 3 parts is (1,1,2) and (1,2,2)... each one you have 3 choices on where to place the 2 and 1 respectively, so 3*2=6 possible ways.